Question

Find the critical numbers of the function. (enter your answers as a comma-separated list. if an answer does not exist, enter dne. ) f(x) = x4/5(x − 8)2

Answer 1

The critical values of the function are x = 0 and x = 64

The critical values of a function f(x) are the values of x for which f'(x) = 0.

Given,

f(x) = $\frac{x^{4} }{5(x-8)^{2} }$

The derivative is found as follows, applying the quotient rule:

f'(x) = $\frac{[5(x^{4} )(x-8)^{2}-5x^{4} [(x-8)^{2} ]' }{[5(x-8)^{2} ]^{2} }$

     = $\frac{2x^{3}(x-64) }{5(x-8)^{3} }$

The zeros of the function are the zeros of the numerator, thus:

$2x^{3} (x-64) =0\\2x^{3} =0-------- > x =0$

x - 64 = 0------------->x = 64

The critical values are x = 0 and x = 64

Learn more about critical values here:brainly.com/question/12958088

#SPJ4