Consider the function f(x)=7(x-4)^2/3

Question

Consider the function f(x)=7(x-4)^2/3

This function has a critical number of A=______

For each of the following intervals, tell whether f'(x) is positive (type in +) or negative (type in -).

(-infinity, A) _____

(A, infinity)______

Thus f(x) has a local ____ at A (type in MAX or MIN)

Answers go in (______)

Answer 1

The critical number(s) of a functions are values where the first derivative is either 0 or not defined. (Graphically, the tangent line at that point would either be horizontal or vertical.)

The first derivative of 7(x-4)^2/3 is

14/3 * (x-4)^-1/3 or 14 / 3(x-4)^1/3

This function is not defined at x = 4 since this would lead to division by zero. So 4 is a critical point.

Checking values of the original function on either side of x=4 yields positive values, so the point in question is a local minimum.