Consider the following function. f(x) = x1/3 + 8 (a) Find the critical numbers o

Question

Consider the following function. f(x) = x1/3 + 8 (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) x = (b) Find the open intervals on which the function is increasing or decreasing. Increasing: (?, 0) (0, ) (?, ) (8, ) (?, 8) none of the above Decreasing: (?, 0) (0, ) (?, ) (8, ) (?, 8) none of the above (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) relative maximum (x, y) = ( ) relative minimum (x, y) = ( ) (d) Use a graphing utility to confirm your results.

Explanation / Answer

f'(x)=1/3x^(-2/3)

a)x=0 is a critical number

because f' is undefined at 0

b) f'(x)>0 for any x so it is increasing on (-,)

decreasing: none of the above

c)relative maximum DNE

relative minimum DNE

because f is always increasing

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