Find the extreme values of the function and where they occur.
-$$y = ( x + 1 ) ^ { 2 / 3 }$$
A) The maximum value is 0 at x = 1.
B) There are no definable extrema.
C) The minimum value is 0 at x = -1.
D) The minimum value is 0 at x = 1.

Question

Quizplus · Accepted Answer

We can find the extreme values of the function by taking the derivative and setting it equal to zero:
$$y'=\frac{2}{3}(x+1)^{-1/3}$$
Setting $y'=0$, we get:
$$\frac{2}{3}(x+1)^{-1/3}=0$$
This equation has no solution, which means that there are no critical points and therefore no definable extrema.

However, we can see that the function is always greater than or equal to 0, since we are taking the cube root of a squared quantity, and any real number squared is non-negative. The only point at which the function is equal to 0 is when $x=-1$, so the minimum value of the function is 0 at $x=-1$. Therefore, the answer is C.

Find the Extreme Values of the Function and Where They y=(x+1)2/3y = ( x + 1 ) ^ { 2 / 3 }y=(x+1)2/3

Find the Extreme Values of the Function and Where They $y = ( x + 1 ) ^ { 2 / 3 }$