c = The input value where the absolute maximum occurs
x = Any input value in the domain of f
Worked Example
Problem: Find the absolute maximum of f(x) = -x² + 4x + 1 on the closed interval [0, 5].
Step 1: Find the derivative and set it equal to zero to locate critical points.
f′(x)=−2x+4=0⟹x=2
Step 2: Confirm that x = 2 lies within the interval [0, 5]. It does.
0≤2≤5✓
Step 3: Evaluate f at the critical point and at both endpoints of the interval.
f(0)=−0+0+1=1
Step 4: Evaluate f at the critical point x = 2.
f(2)=−(4)+8+1=5
Step 5: Evaluate f at the right endpoint x = 5.
f(5)=−(25)+20+1=−4
Step 6: Compare all three values: 1, 5, and −4. The largest is 5, which occurs at x = 2.
max{1,5,−4}=5
Answer: The absolute maximum is 5, occurring at x = 2.
Another Example
Problem: Find the absolute maximum of g(x) = 3x − x³ on the closed interval [−2, 2].
Step 1: Differentiate and find critical points.
g′(x)=3−3x2=0⟹x2=1⟹x=±1
Step 2: Both x = −1 and x = 1 lie in [−2, 2]. Evaluate g at these critical points and at the endpoints.
g(−2)=−6−(−8)=2
Step 3: Evaluate at x = −1.
g(−1)=−3−(−1)=−2
Step 4: Evaluate at x = 1.
g(1)=3−1=2
Step 5: Evaluate at x = 2.
g(2)=6−8=−2
Step 6: Compare all values: 2, −2, 2, −2. The largest value is 2, and it occurs at both x = −2 and x = 1.
max{2,−2,2,−2}=2
Answer: The absolute maximum is 2. It occurs at two points: x = −2 and x = 1. This shows that an absolute maximum value can be achieved at more than one input.
Frequently Asked Questions
Can a function have no absolute maximum?
Yes. If the domain is not a closed interval, the function may increase without bound or approach a value it never reaches. For example, f(x) = x has no absolute maximum on the open interval (0, 10) because it gets arbitrarily close to 10 but never equals it. However, the Extreme Value Theorem guarantees that a continuous function on a closed interval [a, b] always has an absolute maximum.
What is the difference between absolute maximum and relative maximum?
A relative (local) maximum is the highest value in some small neighborhood around a point — the function is higher there than at nearby points. An absolute (global) maximum is the single highest value over the entire domain. Every absolute maximum is also a relative maximum, but a relative maximum is not necessarily the absolute maximum because the function could reach a higher value somewhere else.
Absolute Maximum vs. Relative Maximum
An absolute maximum is the highest function value across the entire domain. A relative maximum is the highest value only within a small surrounding interval. A function can have many relative maxima, but at most one absolute maximum value (though it may occur at multiple points). On a closed interval, the absolute maximum is found by comparing all relative maxima and the endpoint values.
Why It Matters
Finding absolute maxima is essential in optimization problems, where you need to maximize profit, area, efficiency, or other quantities. In physics, the absolute maximum of a projectile's height function tells you the peak altitude. The Closed Interval Method — evaluating at critical points and endpoints — is one of the most practical techniques you learn in calculus.
Common Mistakes
Mistake: Checking only critical points and ignoring the endpoints of a closed interval.
Correction: On a closed interval [a, b], the absolute maximum can occur at an endpoint. Always evaluate the function at both endpoints and at every critical point, then compare all values.
Mistake: Confusing the absolute maximum value with the x-value where it occurs.
Correction: The absolute maximum is the y-value (the output). For f(x) = −x² + 4x + 1 on [0, 5], the absolute maximum is 5 (the y-value), not 2 (the x-value). Be precise: say 'the absolute maximum is 5, occurring at x = 2.'
