Question

Find the extreme values of the function f on the interval [ − 5 , 5 ] . If an extreme value does not exist, enter DNE .

f ( x ) = (x-¹⁶)/(x² +16)

Absolute minimum value:

Absolute maximum value:

Answer 1

To find the extreme values of the function f(x) = (x-16)/(x^2 +16) on the interval [-5, 5], we need to find the critical points and endpoints. The absolute minimum value is -1/3 and the absolute maximum value is 1/3.

Step-by-step explanation:

To find the extreme values of the function f(x) = (x-16)/(x^2 +16) on the interval [-5, 5], we need to find the critical points and endpoints. First, let's find the critical points by taking the derivative of f(x) and solving for x when f'(x) = 0. Taking the derivative of f(x) using the quotient rule, we get f'(x) = (2x(x-16) - (x^2 + 16)(1))/(x^2 + 16)^2. Setting f'(x) = 0 and solving for x, we find x = 0 is a critical point.

Next, we check the endpoints of the interval. Evaluating f(-5) and f(5), we find that f(-5) = -1/3 and f(5) = 1/3.

Therefore, the absolute minimum value is -1/3 and the absolute maximum value is 1/3.