The Central Idea
Consider
\[ f(x)=(x-3)^2+2. \]Because a square is never negative,
\[ (x-3)^2\ge0. \]Therefore,
\[ f(x)\ge2. \]The least output the function can produce is
\[ 2, \]and it occurs when
\[ x=3. \]So:
- the minimum value is \(2\);
- the minimum occurs at \(x=3\);
- the corresponding point is \((3,2)\).
Keeping the value, the input, and the point distinct is important.
Maximum Value
A function has a maximum value \(M\) on a domain if
\[ f(x)\le M \]for every allowed input \(x\), and the function actually reaches \(M\).
That means there is at least one input \(a\) such that
\[ f(a)=M. \]Minimum Value
A function has a minimum value \(m\) on a domain if
\[ f(x)\ge m \]for every allowed input \(x\), and the function actually reaches \(m\).
So there is at least one input \(a\) such that
\[ f(a)=m. \]Value Versus Location
Suppose
\[ f(4)=10 \]and \(10\) is the largest output of the function.
Then:
- the maximum value is \(10\);
- the maximum occurs at \(x=4\);
- the maximum point is \((4,10)\).
These are related but not interchangeable statements.
Absolute Maximum and Minimum
An absolute maximum is the greatest output over the entire domain under consideration.
An absolute minimum is the least output over the entire domain.
They are also called global extrema.
Local Maximum
A local maximum occurs where a function value is at least as large as nearby function values.
Informally, the graph forms a nearby high point.
A local maximum does not have to be the highest point on the entire graph.
Local Minimum
A local minimum occurs where a function value is at least as small as nearby function values.
Informally, the graph forms a nearby low point.
A local minimum need not be the lowest output overall.
Local Versus Absolute
A function can have several local maxima and minima but only one greatest output value and one least output value, if those absolute extrema exist.
Several different inputs can attain the same absolute maximum or minimum value.
A Local Maximum Need Not Be Absolute
Imagine a graph with two hills.
The top of the smaller hill can be a local maximum even if another hill rises higher.
Local comparisons use nearby values; absolute comparisons use the whole domain.
A Local Minimum Need Not Be Absolute
Similarly, a small valley can form a local minimum while another part of the graph lies lower.
The word local means nearby.
Turning Points and Extrema
If a function changes from increasing to decreasing, the transition often creates a local maximum.
If it changes from decreasing to increasing, the transition often creates a local minimum.
This connects extrema directly to increasing and decreasing behavior.
Example: Upward-Opening Parabola
Consider
\[ f(x)=x^2. \]The function decreases on
\[ (-\infty,0) \]and increases on
\[ (0,\infty). \]The minimum occurs at
\[ x=0. \]The minimum value is
\[ \boxed{0}. \]There is no maximum because the function grows without bound.
Example: Downward-Opening Parabola
Consider
\[ f(x)=-x^2+9. \]The greatest output occurs at
\[ x=0. \]Then
\[ f(0)=9. \]So the maximum value is
\[ \boxed{9}. \]There is no minimum over all real numbers because the function decreases without bound.
Vertex Form Makes Quadratic Extrema Visible
For
\[ f(x)=a(x-h)^2+k, \]the vertex is
\[ (h,k). \]If
\[ a>0, \]the parabola opens upward and has minimum value
\[ k. \]If
\[ a<0, \]the parabola opens downward and has maximum value
\[ k. \]Quadratic Example
Let
\[ f(x)=2(x+1)^2-5. \]Because
\[ 2(x+1)^2\ge0, \]we have
\[ f(x)\ge-5. \]So the minimum value is
\[ \boxed{-5}, \]occurring at
\[ \boxed{x=-1}. \]Finding a Quadratic Extremum by Completing the Square
Suppose
\[ f(x)=x^2-6x+11. \]Complete the square:
\[ f(x)=x^2-6x+9+2, \] \[ f(x)=(x-3)^2+2. \]Therefore, the minimum value is
\[ \boxed{2} \]at
\[ \boxed{x=3}. \]Maximum and Minimum from a Table
| \(x\) | \(f(x)\) |
|---|---|
| \(-2\) | 5 |
| 0 | 9 |
| 3 | 4 |
| 7 | 6 |
Among the listed values:
- the maximum output is \(9\);
- the minimum output is \(4\).
The maximum occurs at \(x=0\), and the minimum occurs at \(x=3\).
Tables May Only Show Sampled Extrema
If a table contains selected sample points from a continuous function, the largest listed value is only the largest value in the table.
The actual function may rise higher between sampled inputs.
Use the formula or graph when the problem asks for true extrema over an interval.
Maximum and Minimum from a Graph
On a graph:
- an absolute maximum is the highest attained point;
- an absolute minimum is the lowest attained point;
- local extrema appear as nearby peaks and valleys.
Open circles do not represent attained function values.
Supremum Is Not Always a Maximum
Consider a function with range
\[ (0,5). \]The outputs can get arbitrarily close to \(5\), but never equal \(5\).
So \(5\) is an upper bound, but not a maximum value.
Infimum Is Not Always a Minimum
If the range is
\[ (2,\infty), \]then outputs approach \(2\) but never attain it.
So \(2\) is a lower bound, but not a minimum value.
Maximum Requires Attainment
To call \(M\) a maximum, the function must actually produce \(M\).
Being an upper bound is not enough.
Likewise, a minimum must actually occur.
Open Endpoints Can Prevent Extrema
Consider
\[ f(x)=x \]on the interval
\[ (0,1). \]The outputs approach \(0\) and \(1\), but neither endpoint is included.
So the function has neither a minimum nor a maximum on this domain.
Closed Endpoints Can Create Extrema
Now consider
\[ f(x)=x \]on
\[ [0,1]. \]The minimum value is
\[ 0 \]at \(x=0\), and the maximum value is
\[ 1 \]at \(x=1\).
Endpoints Matter on Restricted Domains
A function that has no global maximum on all real numbers may have one when its domain is restricted.
For example,
\[ f(x)=x^2 \]has no maximum on \(\mathbb{R}\).
But on
\[ [-2,3], \]the maximum value is
\[ 9 \]at \(x=3\).
Restricted-Domain Example
Consider
\[ f(x)=x^2 \]on
\[ [-2,3]. \]The minimum value is
\[ 0 \]at \(x=0\).
The maximum value is
\[ 9 \]at \(x=3\).
Bounded Above
A function is bounded above if there is some number \(M\) such that
\[ f(x)\le M \]for every input in the domain.
The function does not need to attain the smallest possible upper bound.
Bounded Below
A function is bounded below if there is some number \(m\) such that
\[ f(x)\ge m \]for every input in the domain.
Bounded Versus Having an Extremum
A function can be bounded above without having a maximum.
For example,
\[ f(x)=x \]on
\[ (0,1) \]is bounded above by \(1\), but never reaches \(1\).
Functions Can Be Unbounded
The function
\[ f(x)=x \]on all real numbers is unbounded above and unbounded below.
It therefore has neither an absolute maximum nor an absolute minimum.
Maximum and Minimum Can Occur More Than Once
Consider
\[ f(x)=\cos x \]on a domain containing several full cycles.
The maximum value
\[ 1 \]occurs at many inputs.
The minimum value
\[ -1 \]also occurs repeatedly.
The Extremum Value Can Be Unique Even When Its Location Is Not
If a function's largest output is
\[ 10, \]that maximum value is one number.
But the set of inputs where it occurs may contain several points.
Flat Maximum Intervals
A function may attain its maximum across an entire interval.
For example, a piecewise function might satisfy
\[ f(x)=8 \]for
\[ 2\le x\le5 \]and remain below \(8\) elsewhere.
Then every input in \([2,5]\) is a maximizing input.
Flat Minimum Intervals
Similarly, a function may attain its minimum value throughout a flat interval.
Extrema do not have to occur at isolated points.
Piecewise Example
Consider
\[ f(x)= \begin{cases} x+3, & x<0,\\ 3, & 0\le x\le2,\\ 5-x, & x>2. \end{cases} \]The value
\[ 3 \]is attained throughout
\[ [0,2]. \]Depending on the full domain and behavior of the outer pieces, this flat region may form a local or absolute maximum.
Local Extrema and Endpoints
Different courses use slightly different conventions about whether endpoints can count as local extrema.
For absolute extrema, endpoints definitely matter when they belong to the domain.
For local extrema, always follow the stated definition or context.
Local Extrema Without Turning
Some functions can have local extrema without a smooth rounded turning point.
For example,
\[ f(x)=|x| \]has a local and absolute minimum at
\[ x=0, \]even though the graph has a sharp corner there.
Discontinuities Can Affect Extrema
A function may have a jump and still attain a maximum or minimum.
Continuity is not required merely to define extrema.
However, continuity becomes important in theorems that guarantee extrema exist.
Extreme Value Theorem Preview
A major theorem in calculus says that a continuous real-valued function on a closed bounded interval
\[ [a,b] \]must attain both an absolute maximum and an absolute minimum.
This is the Extreme Value Theorem.
Why the Conditions Matter
The theorem requires:
- continuity;
- a closed interval;
- a bounded interval.
Removing one of these conditions can allow extrema to fail to exist.
Example: Missing Closed Endpoint
The function
\[ f(x)=x \]on
\[ (0,1] \]has maximum value \(1\), but no minimum.
The left endpoint is excluded.
Example: Unbounded Domain
The continuous function
\[ f(x)=x^2 \]on
\[ [0,\infty) \]has minimum \(0\), but no maximum.
The domain is closed in one sense but unbounded.
Example: Discontinuity
A discontinuous function on a closed interval may fail to attain an extremum that a continuous function would be guaranteed to attain.
Continuity prevents certain kinds of missing values and jumps.
Finding Extrema Algebraically
Elementary algebra can reveal extrema when a function has useful structure.
Methods include:
- vertex form;
- completing the square;
- absolute-value structure;
- known bounds such as \(x^2\ge0\);
- domain restrictions;
- comparison of endpoints.
Absolute Value Example
Consider
\[ f(x)=|x-4|+2. \]Because
\[ |x-4|\ge0, \]we have
\[ f(x)\ge2. \]The minimum value is
\[ \boxed{2} \]at
\[ \boxed{x=4}. \]Reciprocal Example
Consider
\[ f(x)=\frac{1}{x} \]on
\[ (0,\infty). \]The function decreases toward zero as \(x\) grows, but never reaches zero.
It also grows without bound near \(x=0\).
So it has neither a maximum nor a minimum on this domain.
A Function May Have Only One Type of Absolute Extremum
For
\[ f(x)=e^x, \]the range is
\[ (0,\infty). \]The function has no maximum and no minimum, even though it is bounded below by zero.
Zero is not attained.
Range and Extrema
If the range is
\[ [m,M], \]then the function has minimum value \(m\) and maximum value \(M\).
If the range is
\[ (m,M), \]then neither endpoint is attained, so neither is an extremum.
Half-Open Range Example
If the range is
\[ [2,7), \]then the minimum value is
\[ 2, \]but there is no maximum.
The function approaches \(7\) without attaining it.
Optimization
Optimization asks for the greatest or least possible value of a quantity under stated conditions.
Examples include:
- maximum profit;
- minimum cost;
- maximum area;
- minimum travel time;
- maximum volume;
- minimum material usage.
Optimization Requires a Domain
Suppose a function models profit.
A mathematically possible maximum at a negative production quantity would not make sense if the practical domain requires
\[ q\ge0. \]Optimization must respect both algebraic and contextual constraints.
Example: Maximum Area with Fixed Perimeter
Suppose a rectangle has perimeter
\[ 20. \]If one side has length \(x\), the other is
\[ 10-x. \]The area is
\[ A(x)=x(10-x). \]So
\[ A(x)=-x^2+10x. \]Complete the square:
\[ A(x)=-(x-5)^2+25. \]The maximum area is
\[ \boxed{25} \]when
\[ \boxed{x=5}. \]The optimal rectangle is a square.
Constraints Create the Function
In optimization, a constraint often lets us rewrite several quantities in terms of one independent variable.
Then the resulting function can be analyzed for a maximum or minimum.
Extrema and Derivatives Preview
Later in calculus, derivatives provide a systematic way to search for local extrema.
Important candidates often occur where
\[ f'(x)=0 \]or where the derivative does not exist.
Endpoints must also be checked when the domain is restricted.
Critical Points Preview
A critical point is typically an interior input where the derivative is zero or undefined.
Critical points are candidates for local extrema, but not every critical point is a maximum or minimum.
A Flat Point Need Not Be an Extremum
Consider
\[ f(x)=x^3. \]At
\[ x=0, \]the tangent is horizontal.
But the function is increasing on both sides.
So \(x=0\) is not a local maximum or minimum.
Sign Change Matters
If a function changes from increasing to decreasing, that supports a local maximum.
If it changes from decreasing to increasing, that supports a local minimum.
This behavior-based viewpoint remains useful even before derivatives are introduced.
Worked Example: Absolute Minimum
Let
\[ f(x)=3(x-2)^2+4. \]Since
\[ 3(x-2)^2\ge0, \]we have
\[ f(x)\ge4. \]Thus, the absolute minimum value is
\[ \boxed{4} \]at
\[ \boxed{x=2}. \]Worked Example: Absolute Maximum
Let
\[ f(x)=-2(x+3)^2+10. \]Since
\[ -2(x+3)^2\le0, \]we have
\[ f(x)\le10. \]Therefore, the absolute maximum value is
\[ \boxed{10} \]at
\[ \boxed{x=-3}. \]Worked Example: Restricted Interval
Let
\[ f(x)=x^2-2x \]on
\[ [0,4]. \]Complete the square:
\[ f(x)=(x-1)^2-1. \]The minimum is
\[ -1 \]at
\[ x=1. \]Check endpoints:
\[ f(0)=0, \] \[ f(4)=8. \]So the absolute maximum is
\[ \boxed{8} \]at
\[ \boxed{x=4}. \]Worked Example: No Maximum
Consider
\[ f(x)=x \]on
\[ [0,5). \]The function has minimum value
\[ 0 \]at \(x=0\).
But it has no maximum because the values approach \(5\) without reaching it.
Worked Example: Repeated Maximum
Suppose a function has range
\[ [-2,6] \]and reaches \(6\) at both
\[ x=1 \]and
\[ x=4. \]The maximum value is
\[ \boxed{6}, \]and it occurs at two different inputs.
A Reliable Extremum Workflow
- Identify the domain.
- Determine whether the problem asks for local or absolute extrema.
- Look for turning points, vertices, structural bounds, or flat intervals.
- Check endpoints when they belong to a restricted domain.
- Distinguish the extremum value from the input where it occurs.
- Verify that the candidate is actually attained.
- Interpret the answer in context when solving an applied problem.
A Reliable Quadratic Workflow
- Rewrite the quadratic in vertex form when useful.
- Identify the vertex \((h,k)\).
- Use the sign of the leading coefficient to determine whether the vertex is a maximum or minimum.
- If the domain is restricted, compare the vertex with relevant endpoints.
- Report both the value and where it occurs.
A Reliable Graph Workflow
- Locate the highest and lowest attained points.
- Distinguish open circles from included points.
- Identify smaller local peaks and valleys.
- Read the corresponding input values.
- Check whether the graph is unbounded in either direction.
Common Errors
Confusing the Extremum Value with Its Input
If the minimum occurs at \((3,2)\), the minimum value is \(2\), not \(3\).
Assuming Every Local Maximum Is Absolute
A nearby high point may not be the highest point overall.
Ignoring Endpoints
On a restricted domain, an absolute extremum may occur at an endpoint.
Calling a Bound a Maximum
A maximum must actually be attained.
Assuming Every Function Has Both Extrema
Unbounded functions and open domains may have one or neither.
Ignoring the Domain
Changing the domain can change which extrema exist.
Assuming a Flat Tangent Guarantees an Extremum
A function can have a horizontal tangent and continue increasing through the point.
Practice
1. Quadratic Minimum
Find the minimum value of
\[ f(x)=(x-4)^2+3. \]Check Answer
The square term is minimized at \(x=4\).
Minimum value:
\[ \boxed{3}. \]It occurs at
\[ \boxed{x=4}. \]2. Quadratic Maximum
Find the maximum value of
\[ f(x)=-3(x+2)^2+7. \]Check Answer
The maximum value is
\[ \boxed{7} \]at
\[ \boxed{x=-2}. \]3. No Maximum
Does
\[ f(x)=x^2 \]have an absolute maximum on \(\mathbb{R}\)?
Check Answer
No. The function is unbounded above.
4. Open Interval
Does
\[ f(x)=x \]on
\[ (0,1) \]have a maximum or minimum?
Check Answer
No. The function approaches both endpoint values but attains neither.
5. Closed Interval
Find the absolute extrema of
\[ f(x)=x \]on
\[ [-2,5]. \]Check Answer
Minimum value:
\[ \boxed{-2} \]at \(x=-2\).
Maximum value:
\[ \boxed{5} \]at \(x=5\).
6. Complete the Square
Find the minimum value of
\[ f(x)=x^2-8x+20. \]Check Answer
\[ f(x)=x^2-8x+16+4, \] \[ f(x)=(x-4)^2+4. \]Minimum value:
\[ \boxed{4} \]at
\[ \boxed{x=4}. \]7. Range and Extrema
A function has range
\[ [-3,8). \]What extrema exist?
Check Answer
The function has minimum value
\[ \boxed{-3}. \]It has no maximum because \(8\) is not attained.
8. Value Versus Location
A graph has its lowest point at
\[ (6,-4). \]State the minimum value and where it occurs.
Check Answer
Minimum value:
\[ \boxed{-4}. \]It occurs at
\[ \boxed{x=6}. \]Key Ideas
- A maximum is the greatest attained output; a minimum is the least attained output.
- The extremum value is different from the input where it occurs.
- Absolute extrema compare the whole domain; local extrema compare nearby values.
- Turning from increasing to decreasing often creates a local maximum.
- Turning from decreasing to increasing often creates a local minimum.
- Quadratic extrema are especially visible in vertex form.
- Endpoints matter when the domain is restricted.
- An upper or lower bound is not automatically a maximum or minimum because the bound may not be attained.
- A function may have one, both, or neither absolute extremum.
- Several inputs may attain the same maximum or minimum value.
- Flat intervals can contain many maximizing or minimizing inputs.
- Optimization problems ask for extrema subject to constraints.
- The Extreme Value Theorem later explains when continuous functions are guaranteed to attain absolute extrema.
- Derivative methods will later provide a systematic way to locate and classify extrema.
Where This Leads
Maximum and minimum values turn qualitative function behavior into optimization questions.
These ideas lead directly into local and global extrema, boundedness, vertex methods, constrained optimization, critical points, derivative tests, the Extreme Value Theorem, and applied problems where the goal is to find the best, largest, smallest, fastest, cheapest, or most efficient possible outcome.