Provide an appropriate response.Let f(x) = (x - 8)2/3 (a) Does f'(8) exist? (b) Show that the only local extreme value of f occurs at x = 8. (c) Does the result of (b) contradict the Extreme Value Theorem? (d) Repeat parts (a) and (b) for f(x) = (x - c)2/3.Give reasons for your answers.

Question

What will be an ideal response?

Answer 1

(a) No, since f'(x) = (x - 8)^-1/3, which is undefined at x = 8.
(b) The derivative is defined and nonzero for all x ? 8.
(c) No, f(x) need not have a global maximum because its domain is all real numbers. Any restriction of f to a closed interval of the form [a, b] would have both a maximum value and a minimum value on the interval.
(d) The answers are the same as (a) and (b) with 8 replaced by c.