Skill: Identify local and absolute extrema

A local extremum compares a function with nearby inputs. An absolute extremum compares the function with every input in its stated domain.

• A local maximum is higher than nearby function values.
• A local minimum is lower than nearby function values.
• An absolute maximum is the greatest value on the entire domain.
• An absolute minimum is the least value on the entire domain.

Worked example

Find the local and absolute extrema of

$$f(x)=x^3-3x^2-9x+5$$

on the domain

$$-2\le x\le 4.$$

Step 1: Find possible turning points

Differentiate:

$$f'(x)=3x^2-6x-9$$

Factor:

$$f'(x)=3(x+1)(x-3).$$

Set the derivative equal to zero:

$$3(x+1)(x-3)=0.$$

Thus, the possible turning points occur at

$$x=-1 \quad \text{and} \quad x=3.$$

Step 2: Decide whether each point is a local maximum or minimum

Check the sign of $f'(x)$:

• For $x<-1$, $f'(x)>0$, so the function is increasing.
• For $-1<x<3$, $f'(x)<0$, so the function is decreasing.
• For $x>3$, $f'(x)>0$, so the function is increasing.

At $x=-1$, the function changes from increasing to decreasing, so there is a local maximum.

At $x=3$, the function changes from decreasing to increasing, so there is a local minimum.

Step 3: Find the function values

Evaluate the function at the turning points and at both domain endpoints:

$$f(-2)=3$$ $$f(-1)=10$$ $$f(3)=-22$$ $$f(4)=-15$$

The values are:

$$\begin{array}{c|c} x & f(x)\\ \hline -2 & 3\\ -1 & 10\\ 3 & -22\\ 4 & -15 \end{array}$$

Step 4: Identify all extrema

• At $x=-1$, $f(-1)=10$ is a local maximum.
• At $x=3$, $f(3)=-22$ is a local minimum.
• The greatest value in the entire domain is $10$, so the absolute maximum is $10$ at $x=-1$.
• The least value in the entire domain is $-22$, so the absolute minimum is $-22$ at $x=3$.

Always check the domain’s endpoints when finding absolute extrema. An endpoint can be the absolute maximum or minimum, even though the main turning points occur inside the interval.