## Calculus AB and Calculus BC

## CHAPTER 4 Applications of Differential Calculus

### F. GLOBAL MAXIMUM OR MINIMUM

CASE I. DIFFERENTIABLE FUNCTIONS.

If a function *f* is differentiable on a closed interval *a* ≤ *x* ≤ *b*, then *f* is also continuous on the closed interval [*a*, *b*] and we know from the Extreme Value Theorem that *f* attains both a (global) maximum and a (global) minimum on [*a*, *b*]. To find these, we solve the equation *f* *′*(*x*) = 0 for critical points on the interval [*a*, *b*], then evaluate *f* at each of those and also at *x* = *a* and *x* = *b*. The largest value of *f* obtained is the global max, and the smallest the global min.

**EXAMPLE 16**

Find the global max and global min of *f* on (a) −2 ≤ *x* ≤ 3, and (b) 0 ≤ *x* ≤ 3, if *f* (*x*) = 2*x*^{3} − 3*x*^{2} − 12*x*.

**SOLUTION:**

**(a)** *f* *′*(*x*) = 6*x*^{2} − 6*x* − 12 = 6(*x* + 1)(*x* − 2), which equals zero if *x* = −1 or 2. Since *f* (−2) = −4, *f* (−1) = 7, *f* (2) = −20, and *f* (3) = −9, the global max of *f* occurs at *x* = −1 and equals 7, and the global min of *f* occurs at *x* = 2 and equals −20.

**(b)** Only the critical value 2 lies in [0,3]. We now evaluate *f* at 0, 2, and 3. Since *f* (0) = 0, *f* (2) = −20, and *f* (3) = −9, the global max of *f* equals 0 and the global min equals −20.

CASE II. FUNCTIONS THAT ARE NOT EVERYWHERE DIFFERENTIABLE.

We proceed as for Case I but now evaluate *f* also at each point in a given interval for which *f* is defined but for which *f* *′* does not exist.

**EXAMPLE 17**

The absolute-value function *f* (*x*) = |*x*| is defined for all real *x*, but *f* *′*(*x*) does not exist at *x* = 0. Since *f* *′*(*x*) = −1 if *x* < 0, but *f* *′*(*x*) = 1 if *x* > 0, we see that *f* has a global min at *x* = 0.

**EXAMPLE 18**

The function has neither a global max nor a global min on *any* interval that contains zero (see Figure N2–4). However, it does attain both a global max and a global min on every closed interval that does not contain zero. For instance, on [2,5] the global max of *f* is the global min