# Consider the function f(x)= 4(x - 3)^{2/3}. For this function there are two important intervals:...

## Question:

Consider the function {eq}f(x)= 4(x - 3)^{2/3} {/eq}. For this function there are two important intervals: {eq}(-\infty, A)\ and\ (A, \infty {/eq}) where A is a critical number. Find A.

## Critical Points:

Let {eq}f(x) {/eq} be a function. The critical points of {eq}f(x) {/eq} are the points {eq}c {/eq} in the domain of {eq}f(x) {/eq} such that {eq}f'(c)=0 {/eq} or {eq}f'(c) {/eq} is undefined. The critical points play an important role in analyzing a graph since they are the places where the function has relative maximum and minimums. That is places where the function may switch from increasing to decreasing or decreasing to increasing

Let {eq}f(x)=4(x-3)^{2/3} {/eq}. We first note that the domain of {eq}f(x) {/eq} is {eq}(-\infty,\infty) {/eq}. Next we note that

{eq}\begin{align} f'(x)&=4\frac{2}{3}(x-3)^{-1/3}\\ &=\dfrac{8}{3(x-3)^{1/3}} \end{align} {/eq}

Next we see that {eq}f'(x)\neq 0 {/eq} for any {eq}x {/eq} and that {eq}f'(x) {/eq} is undefined when the denominator becomes 0 at {eq}x=3 {/eq}. We conclude that the only critical point is {eq}A=3 {/eq}.