Find the extreme values of the function f on the interval [1,5]. If an extreme value does not exist, enter DNE . f(x)=x^4+4/x

Question

Absolute minimum value=

Absolute maximum value=

Please help!!!

Answer 1

On C¹ functions defined over closed intervals, you check the function value where the first derivative equals 0 and at the endpoints.

f(x) = x⁴ + 4/x

f'(x) = 4x³ - 4/x² = 4/x² (x⁵ - 1)

This is only 0 at the endpoint, 1. Thus, we only look at our endpoints.

f(1) = 1⁴ + 4/1 = 1 + 4 = 5

f(5) = 5⁴ + 4/5 = 625 4/5

The minimum is 5 at x = 1; the maximum is 625 4/5 at x = 5.