Find the extreme values of the function and where they occur.
-$$y = ( x - 5 ) ^ { 2 / 3 }$$
A) The minimum value is 0 at x = -5.
B) The maximum value is 0 at x = -5.
C) There are no definable extrema.
D) The minimum value is 0 at x = 5.

Question

Quizplus · Accepted Answer

To find the extreme values, we must take the derivative of the function and set it equal to zero:

$$y' = \frac{2}{3}(x-5)^{-1/3}$$

Setting y' equal to zero, we get:

$$0 = \frac{2}{3}(x-5)^{-1/3}$$

This is only true if the numerator is equal to zero, so we have:

$$0 = 2$$
which is a contradiction. Therefore, there is no critical point and no definable extrema. However, we can see that the function is always non-negative since it is the cube root of a squared quantity. Therefore, the minimum value is 0, which occurs at x = 5.

Find the Extreme Values of the Function and Where They y=(x−5)2/3y = ( x - 5 ) ^ { 2 / 3 }y=(x−5)2/3

Find the Extreme Values of the Function and Where They $y = ( x - 5 ) ^ { 2 / 3 }$