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Maxima and Minima of Functions — Definition, Formula & Examples

Maxima and minima are the highest and lowest values a function reaches, either on its entire domain or within a specific interval. A maximum is a peak and a minimum is a valley on the graph of a function.

A function ff has a local maximum at x=cx = c if f(c)≥f(x)f(c) \geq f(x) for all xx in some open interval containing cc, and a local minimum at x=cx = c if f(c)≤f(x)f(c) \leq f(x) for all xx in some open interval containing cc. An absolute (global) maximum or minimum is the largest or smallest value of ff on its entire domain or on a closed interval.

How It Works

To find maxima and minima, start by computing the first derivative f′(x)f'(x) and setting it equal to zero to locate critical numbers. Check endpoints of the interval if one is given. Use the first derivative test or second derivative test to classify each critical number as a local maximum, local minimum, or neither. On a closed interval [a,b][a, b], the absolute maximum and minimum must occur at critical numbers or at the endpoints.

Worked Example

Problem: Find all local maxima and minima of f(x)=x3−3xf(x) = x^3 - 3x on the interval [−2,2][-2, 2].
Find the derivative: Differentiate the function.
f′(x)=3x2−3f'(x) = 3x^2 - 3
Find critical numbers: Set the derivative equal to zero and solve.
3x2−3=0  ⟹  x2=1  ⟹  x=−1 or x=13x^2 - 3 = 0 \implies x^2 = 1 \implies x = -1 \text{ or } x = 1
Evaluate at critical numbers and endpoints: Compute ff at x=−2,−1,1,2x = -2, -1, 1, 2.
f(−2)=−2,f(−1)=2,f(1)=−2,f(2)=2f(-2) = -2,\quad f(-1) = 2,\quad f(1) = -2,\quad f(2) = 2
Answer: The absolute maximum value is 22, occurring at x=−1x = -1 and x=2x = 2. The absolute minimum value is −2-2, occurring at x=1x = 1 and x=−2x = -2. Locally, x=−1x = -1 is a local maximum and x=1x = 1 is a local minimum.

Why It Matters

Optimization problems in physics, engineering, and economics all reduce to finding maxima or minima. For example, maximizing profit, minimizing material cost, or finding the highest point a projectile reaches each requires this technique. It is also central to the AP Calculus AB and BC exams.

Common Mistakes

Mistake: Forgetting to check the endpoints of a closed interval.
Correction: On a closed interval [a,b][a, b], the absolute max or min can occur at an endpoint, not just at a critical number. Always evaluate ff at both endpoints.