Using the first and second derivative to reveal function behavior — where functions rise, fall, curve, peak, and bend — without point-by-point plotting
Derivatives do more than measure rates of change at a single point. When you know how a function's derivative behaves across an interval — where it is positive, where it is negative, where it equals zero — you know the shape of the function itself. This chapter teaches you to use derivatives as tools for reading the behavior of functions: where they rise and fall, where they reach peaks and valleys, how they curve, and where their concavity changes.
This kind of analysis turns algebraic information into geometric understanding. You will learn to construct sign charts for the first and second derivative, identify critical numbers and classify them as local maxima, local minima, or neither, determine where a function is concave up or concave down, find inflection points, and use all of this information together to sketch accurate graphs without relying on point-by-point plotting.
By the end of this chapter, you will understand that the derivative is not just a calculation — it is a language for describing function behavior.
Suppose you are analyzing the path of a roller coaster. You want to know where the track rises and where it falls, where the highest and lowest points are, where the ride curves upward like a bowl and where it curves downward like a hill, and where the curvature transitions from one type to the other. You are not asked to evaluate the height at every point. You are asked to understand the shape of the ride. Derivatives give you exactly this kind of structural information. The first derivative tells you whether the height is increasing or decreasing. The second derivative tells you whether the curvature is concave up or concave down. Their combined information tells you the story of the shape.
Before diving into formal rules and procedures, it is worth pausing to think about what derivatives actually reveal about a function's graph. Recall that the derivative f′(x) represents the instantaneous rate of change of f at the point x. If you think of the graph of f as a road you are driving along, then the derivative at each point tells you the slope of the road at that moment. When the slope is positive, you are going uphill. When the slope is negative, you are going downhill. When the slope is zero, you are momentarily on level ground — at a peak, a valley, or a flat section.
This connection between derivative sign and graph direction is one of the most useful insights in all of calculus. Instead of computing function values at dozens of points to understand a graph, you can compute the derivative and analyze its sign across intervals.
If f′(x) > 0 on an interval, then f is increasing on that interval.
If f′(x) < 0 on an interval, then f is decreasing on that interval.
If f′(x) = 0 at a point, then f has a horizontal tangent there — a candidate for a local extremum.
The second derivative adds another layer of information. If f″(x) > 0 on an interval, the graph is concave up — it curves upward like the inside of a bowl. If f″(x) < 0 on an interval, the graph is concave down — it curves downward like the outside of a hill. Where concavity changes from one type to the other, the graph has an inflection point. The combination of first and second derivative information gives a complete portrait of a function's behavior without needing to evaluate the function at every point.
Draw a smooth curve that rises from left to right, reaches a peak, falls to a valley, then rises again. Mark the rising intervals with a "+" label for f′, the falling interval with a "–" label for f′, and place arrows on the curve showing the direction of travel. This is a simple visual version of the relationship between derivative sign and graph direction.
A function f is increasing on an interval (a, b) if, for every pair of values x₁ and x₂ in that interval with x₁ < x₂, it follows that f(x₁) < f(x₂). In other words, as x moves to the right, f(x) moves upward. A function f is decreasing on (a, b) if f(x₁) > f(x₂) whenever x₁ < x₂ — as x moves right, f(x) moves down.
Let f be differentiable on an open interval (a, b).
If f′(x) > 0 for all x in (a, b), then f is increasing on (a, b).
If f′(x) < 0 for all x in (a, b), then f is decreasing on (a, b).
Organizing the sign of f′ across intervals in a sign chart is the clearest way to track behavior across the whole domain. You will use sign charts repeatedly throughout this chapter and the next.
f′(x) = 3x² − 6x − 9
Set f′(x) = 0:
3x² − 6x − 9 = 0
x² − 2x − 3 = 0
(x − 3)(x + 1) = 0
x = 3 or x = −1
These values divide the number line into three intervals: (−∞, −1), (−1, 3), and (3, ∞).
On (−∞, −1): test x = −2
f′(−2) = 3(4) − 6(−2) − 9 = 12 + 12 − 9 = 15 > 0 → increasing
On (−1, 3): test x = 0
f′(0) = 0 − 0 − 9 = −9 < 0 → decreasing
On (3, ∞): test x = 4
f′(4) = 3(16) − 6(4) − 9 = 48 − 24 − 9 = 15 > 0 → increasing
x: ... −1 ... 3 ...
f′(x): + 0 − 0 +
f(x): ↑ ↓ ↑
A value x = c is a critical number of f if c is in the domain of f and either:
f′(c) = 0 — the tangent line is horizontal at x = c, or
f′(c) does not exist — a corner, cusp, or vertical tangent at x = c.
Critical numbers are the candidates for local extrema — the peaks and valleys of the graph. The word candidate matters here: not every critical number is a local maximum or minimum. Some are inflection points with horizontal tangents, or corners, or cusps. But every local maximum and every local minimum of a differentiable function occurs at a critical number.
Critical numbers are x-values (inputs), not y-values. The critical point is the full coordinate pair (c, f(c)). Always compute both.
g′(x) = 4x³ − 12x² = 4x²(x − 3)
Set g′(x) = 0:
4x²(x − 3) = 0
x = 0 or x = 3
Since g′(x) is a polynomial, it exists everywhere. There are no points where the derivative fails to exist.
A function f has a local maximum at x = c if f(c) ≥ f(x) for all x in some open interval around c. It has a local minimum at x = c if f(c) ≤ f(x) for all x in some open interval around c. Together, local maxima and minima are called local extrema.
Local extrema are not the same as absolute extrema. A local maximum might not be the highest point on the entire graph — it is only the highest point near its location. Absolute extrema are studied more carefully in Chapter 8.
Near a local maximum, the function must first rise, reach the peak, then fall. This means f′ must transition from positive to negative at the critical number. Near a local minimum, the function must first fall, reach the valley, then rise — f′ transitions from negative to positive. This observation is the key idea behind the First Derivative Test.
Suppose x = c is a critical number of a continuous function f.
If f′(x) changes from + to − at x = c, then f has a local maximum at c.
If f′(x) changes from − to + at x = c, then f has a local minimum at c.
If f′(x) does not change sign at x = c (remains positive or negative on both sides), then f has neither a local maximum nor a local minimum at c.
x: ... −1 ... 3 ...
f′(x): + 0 − 0 +
f(−1) = (−1)³ − 3(−1)² − 9(−1) + 5 = −1 − 3 + 9 + 5 = 10
Local maximum point: (−1, 10).
f(3) = 27 − 27 − 27 + 5 = −22
Local minimum point: (3, −22).
h′(x) = 3x²
Set h′(x) = 0: 3x² = 0 → x = 0
x: ... 0 ...
h′(x): + 0 +
h′(x) = 3x² ≥ 0 for all x. The sign does not change at x = 0.
Do not assume that every critical number is a local maximum or minimum. Example 7.5.2 shows this is wrong. A sign chart is needed to determine the correct classification every time.
The first derivative tells you whether a function is increasing or decreasing. The second derivative tells you something deeper: how the rate of change itself is changing. This is called concavity.
The graph of f is concave up on an interval if f″(x) > 0 for all x on that interval. The graph curves upward like the inside of a bowl. Tangent lines lie below the curve.
The graph of f is concave down on an interval if f″(x) < 0 for all x on that interval. The graph curves downward like the outside of a hill. Tangent lines lie above the curve.
A function can be increasing and concave up; increasing and concave down; decreasing and concave up; or decreasing and concave down. Concavity is about how the slope is changing, not about whether the function is rising or falling.
| Direction of f | Concavity | Meaning | Example |
|---|---|---|---|
| Increasing | Concave Up | Slope positive and growing (accelerating upward) | Right half of y = x² |
| Increasing | Concave Down | Slope positive but shrinking (decelerating upward) | y = √x for x > 0 |
| Decreasing | Concave Up | Slope negative but becoming less negative (decelerating downward) | Left half of y = x² |
| Decreasing | Concave Down | Slope negative and growing more negative (accelerating downward) | y = −ex |
f′(x) = 3x² − 6x − 9
f″(x) = 6x − 6
Set f″(x) = 0: 6x − 6 = 0 → x = 1
On (−∞, 1): test x = 0 → f″(0) = −6 < 0 → concave down
On (1, ∞): test x = 2 → f″(2) = 6 > 0 → concave up
A point (c, f(c)) on the graph of f is an inflection point if f is continuous at c and the concavity changes — from concave up to concave down, or vice versa — at c.
A value where f″(c) = 0 is only a candidate for an inflection point. If the concavity does not actually change there, it is not an inflection point. Always verify the sign change.
On (−∞, 1): f″(x) < 0 (concave down)
On (1, ∞): f″(x) > 0 (concave up) ✓ Concavity changes at x = 1.
f(1) = 1 − 3 − 9 + 5 = −6
p′(x) = 4x³
p″(x) = 12x²
Set p″(x) = 0: 12x² = 0 → x = 0
Suppose f′(c) = 0 and f″(c) exists.
If f″(c) > 0, then f has a local minimum at c. (Concave up at that point — bottom of a bowl.)
If f″(c) < 0, then f has a local maximum at c. (Concave down at that point — top of a hill.)
If f″(c) = 0, the test is inconclusive. Use the First Derivative Test instead.
f′(x) = 6x² − 6x − 12 = 6(x² − x − 2) = 6(x − 2)(x + 1)
Critical numbers: x = 2 and x = −1
f″(x) = 12x − 6
f″(2) = 12(2) − 6 = 18 > 0 → local minimum at x = 2
f″(−1) = 12(−1) − 6 = −18 < 0 → local maximum at x = −1
f(2) = 16 − 12 − 24 + 4 = −16
f(−1) = −2 − 3 + 12 + 4 = 11
Use the First Derivative Test when f″(c) = 0 at a critical number, when f″ is difficult to compute, or when you are already constructing a sign chart for f′. Use the Second Derivative Test when f″ is easy to compute and evaluate, and you want a quick classification at an isolated critical number.
With all the pieces in place — domain, intercepts, increasing/decreasing intervals, critical points and their classifications, concavity intervals, and inflection points — you now have enough information to sketch a fairly accurate graph of a function without plotting dozens of points.
1. Domain — Identify the domain. Are there restrictions from square roots, denominators, logarithms?
2. Intercepts — Find f(0) for the y-intercept. Solve f(x) = 0 for x-intercepts if practical.
3. Asymptotes — Vertical asymptotes where denominators are zero; horizontal asymptotes using limits at infinity; end behavior for polynomials.
4. First derivative — Find f′(x). Build a sign chart. Identify increasing/decreasing intervals and critical numbers.
5. Local extrema — Classify critical numbers. Compute f-values at each critical number.
6. Second derivative — Find f″(x). Build a sign chart. Identify concavity intervals.
7. Inflection points — Confirm sign changes in f″. Compute f-values at each inflection point.
8. Sketch — Plot intercepts, extrema, and inflection points. Connect with correct direction and concavity. Add asymptotes.
f′(x) = 3(x − 3)(x + 1) → critical numbers: x = −1, x = 3
Sign chart:
x: (−∞, −1) −1 (−1, 3) 3 (3, ∞)
f′: + 0 − 0 +
f: ↑ ↓ ↑
Local maximum at x = −1: f(−1) = 10 → point (−1, 10)
Local minimum at x = 3: f(3) = −22 → point (3, −22)
f″(x) = 6x − 6 → candidate: x = 1
Sign chart:
x: (−∞, 1) 1 (1, ∞)
f″: − 0 +
concave ↓ concave ↑
Concavity changes at x = 1.
f(1) = 1 − 3 − 9 + 5 = −6 → inflection point (1, −6)
Plot these key points: local max at (−1, 10), y-intercept at (0, 5), inflection point at (1, −6), and local min at (3, −22). Coming from the left, the curve rises steeply (concave down as it approaches the local max), reaches the peak at (−1, 10), curves through the inflection at (1, −6) where it transitions to concave up, then falls to the minimum at (3, −22), and rises steeply thereafter. The overall shape is the typical S-shaped cubic.
One of the richest skills in calculus is reading information about f, f′, and f″ simultaneously — from separate graphs, from a table, or by reasoning through the relationships. The following table summarizes the connections between all three.
| Condition on f | Condition on f′ | Condition on f″ |
|---|---|---|
| f is increasing | f′(x) > 0 | — |
| f is decreasing | f′(x) < 0 | — |
| f has a local maximum | f′(c) = 0, sign changes + → − | f″(c) < 0 (if SDT applies) |
| f has a local minimum | f′(c) = 0, sign changes − → + | f″(c) > 0 (if SDT applies) |
| f is concave up | f′ is increasing | f″(x) > 0 |
| f is concave down | f′ is decreasing | f″(x) < 0 |
| f has an inflection point | f′ has a local extremum at c | f″(c) = 0 and sign changes |
Draw three curves vertically aligned. The top curve is a smooth cubic f. The middle curve is its derivative f′ — a parabola opening upward with roots matching the critical numbers of f. The bottom curve is f″ — a line. Mark clearly where f′ = 0 corresponds to extrema on f, and where f″ = 0 corresponds to the inflection point of f and the vertex of f′. This three-panel diagram is one of the most instructive exercises in first-year calculus.
f′(x) > 0 on (−∞, 2) — f is increasing
f′(2) = 0 — local maximum
f′(x) < 0 on (2, 5) — f is decreasing
f′(5) = 0 — local minimum
f′(x) > 0 on (5, ∞) — f is increasing again
f″(x) < 0 on (−∞, 3.5) — concave down
f″(3.5) = 0 — inflection point candidate
f″(x) > 0 on (3.5, ∞) — concave up
Also: since f transitions from concave down to concave up at x = 3.5, f′ has a local minimum at x = 3.5.
Graphing calculators and computer algebra systems can confirm your algebraic analysis and help you visualize functions that are too complex to sketch by hand. However, technology is a verification tool, not a replacement for understanding.
Setting a good window. Before you trust a graph, make sure the window shows all relevant features. Include the domain of interest, the critical points, the inflection points, any asymptotes, and enough end behavior to see the overall shape. A default window from −10 to 10 may miss important features entirely.
Zooming to verify local behavior. Near a critical point, zoom in to see whether the function actually reaches a local max or min there. Near a potential inflection point, zoom in to confirm the curvature change.
Comparing your sketch to the technology graph. Before graphing on a calculator, produce your own sketch using the derivative analysis from Sections 7.2 through 7.9. Then compare. If they agree, your analysis is confirmed. If they disagree, find the error in your algebra or your reasoning.
If the calculator graph shows a local maximum where your analysis predicts a local minimum, go back and check the sign chart. Either the derivative was computed incorrectly, or the test values were substituted wrong. Technology confirms your algebra; it does not replace the need for correct algebra.
f′(x) = cos x − 1
Now, cos x − 1 ≤ 0 for all x, and cos x − 1 = 0 only when cos x = 1, i.e., x = 0, 2π, 4π, …
Graph analysis with derivatives is not purely abstract. The same ideas describe meaningful behavior of real quantities across many fields.
In motion. If s(t) is the position of an object and s′(t) = v(t) is its velocity, then v(t) > 0 means the object moves in the positive direction, v(t) < 0 means it moves in the negative direction, and v(t) = 0 is a critical number where the object momentarily stops. The acceleration a(t) = v′(t) plays the role of f″ — where a(t) > 0, the velocity is increasing.
In economics. If P(x) is the profit from producing x units, P′(x) > 0 means profit is increasing as production increases, and P′(x) = 0 at a critical number — possibly the production level that maximizes profit. The condition P″(x) < 0 at a critical number confirms a local maximum profit by the Second Derivative Test.
In biology. If N(t) is a population at time t, N′(t) > 0 means the population is growing. An inflection point in N(t) corresponds to when the rate of growth is at its maximum — a critical moment in epidemiology and ecology. After the inflection point, the population still grows, but more slowly.
In data science and machine learning. Optimization algorithms that train machine learning models search for local minima in a loss function. Gradient descent uses information analogous to f′(x) — the gradient — to move in the direction of steeper descent. Understanding concavity helps practitioners understand whether their algorithms are converging.
Set N(t) = 500:
1000 / (1 + 9e^{−0.5t}) = 500
1 + 9e^{−0.5t} = 2
9e^{−0.5t} = 1
e^{−0.5t} = 1/9
−0.5t = ln(1/9) = −ln 9
t = 2 ln 9 ≈ 4.39 years
Increasing/decreasing depends on f′. Concavity depends on f″. A function can be increasing and concave down (slopes are positive but getting smaller). All four combinations — increasing/concave up, increasing/concave down, decreasing/concave up, decreasing/concave down — are possible and all occur in real functions.
Example 7.5.2 showed that f(x) = x³ has a critical number at x = 0 that is neither a local max nor a local min. Always check for a sign change in f′ before classifying.
Example 7.7.2 showed that f(x) = x⁴ has f″(0) = 0 but no inflection point at x = 0 — the concavity does not change. Always check whether f″ actually changes sign at the candidate.
Inconclusive means the test cannot decide — not that the answer is "neither." When f″(c) = 0, switch to the First Derivative Test to make a correct classification.
Identifying the x-value is only half the work. The actual point on the graph requires both the x-coordinate and f(x). Always compute f(c) at critical numbers and inflection point candidates.
Each test value must be strictly inside its interval, not at its boundary. Choosing a critical number as a test value gives f′(c) = 0, which does not reveal the sign on the interval.
If f(x) = √(x − 2), the domain is [2, ∞). Critical number analysis and sign charts must stay within the domain. Values outside the domain are irrelevant.
A local maximum is the highest point locally — near c. An absolute maximum is the highest point on the entire domain. A local maximum can be well below an absolute maximum elsewhere. Chapter 8 treats absolute extrema fully.
A local max of f′ is an inflection point of f, not a local max of f. When a graph of f′ is given, translate carefully: peaks and valleys of f′ become inflection points of f; zeros of f′ become critical numbers of f.
A local minimum of f′ at c means f′(c) is the smallest slope value nearby — but f′(c) may still be positive, meaning f is still increasing through that inflection point. The sign of f′ determines direction; the derivative of f′ determines concavity.
Chapter 7 established all the derivative tools needed to understand function behavior: increasing and decreasing intervals, critical points, local extrema, concavity, and inflection points. Chapter 8 (Optimization and Applied Derivative Models) will use these tools to solve applied maximum and minimum problems — finding the largest area, the shortest distance, the maximum profit, the minimum cost, the most efficient design.
The transition from Chapter 7 to Chapter 8 is from analyzing function behavior to using that behavior to answer real questions about the best or worst value of a quantity. To prepare, keep these connections in mind:
Critical numbers are where extrema can occur. In Chapter 8, you will also need to check the endpoints of a closed interval (the Closed Interval Method). The Second Derivative Test will help confirm whether a critical number gives a maximum or minimum in physical models. Units will return with full force — when the function being optimized represents cost in dollars, area in square meters, or volume in cubic centimeters, every critical number has a physical meaning that must be interpreted.
The derivative techniques of Chapters 3 through 7 are the foundation for everything in applied mathematics that involves maximizing or minimizing a quantity. The optimization problems in Chapter 8 connect directly to engineering design, economic profit analysis, physics energy minimization, machine learning loss minimization, and scientific computing.
For each function, find the critical numbers, build a sign chart for f′(x), and determine increasing and decreasing intervals.
Find critical numbers and classify each as local maximum, local minimum, or neither using the First Derivative Test.
Find the intervals of concavity and any inflection points.
Find critical numbers and classify them using the SDT where possible. If inconclusive, use the First Derivative Test.
Use the full curve sketching strategy to sketch each function. Label all critical points, inflection points, and any asymptotes.
| x | f′(x) |
|---|---|
| 0 | −3 |
| 1 | −1 |
| 2 | 0 |
| 3 | 2 |
| 4 | 4 |
| 5 | 3 |
| 6 | 0 |
| 7 | −2 |
Each problem shows a student's work. Identify the error and write the correct solution.
Identify the logical error. The conclusion may be correct, but the reasoning is flawed. What should the student have done instead of citing the Second Derivative Test result?
Identify the error and state the correct conclusion. What step did this student skip?
What is wrong with this reasoning? What should the student do?
Explain the error. A local maximum of f′ at x = 4 corresponds to what feature on the graph of f?
For f(x) = x³ − 6x² + 9x + 2:
Chapter 7 gave you the tools to understand function behavior: where functions rise, fall, peak, and curve. Chapter 8 will use these tools to answer a more pointed question — what is the best value? Finding the largest area, the shortest distance, the maximum profit, the minimum cost, the most efficient design — all of these are optimization problems. You will apply the critical number machinery from this chapter to real-world objective functions, learn the Closed Interval Method for finding absolute extrema on bounded domains, and practice reading, representing, solving, checking, and explaining every answer in context with correct units.