Question

use a graphing utility to graph the function. use a -5.5,1 by -5.5,1 viewing rectangle. then find the intervals on which the function is increasing, decreasing, or constant. g(x)=x^{6/11} choose the correct graph below.

Explanation:

Step1: Analyze the function $g(x)=x^{\frac{6}{11}}=\sqrt[11]{x^{6}}$.

The function is defined for all real - numbers since we can take the 11th root of any real number and then raise it to the 6th power. The function is an even function because $g(-x)=(-x)^{\frac{6}{11}}=\sqrt[11]{(-x)^{6}}=\sqrt[11]{x^{6}} = g(x)$.

Step2: Consider the derivative to find intervals of increase and decrease.

First, use the power rule $y = x^{n}\Rightarrow y^\prime=nx^{n - 1}$. For $y = x^{\frac{6}{11}}$, $y^\prime=\frac{6}{11}x^{\frac{6}{11}-1}=\frac{6}{11}x^{-\frac{5}{11}}=\frac{6}{11x^{\frac{5}{11}}}$. The derivative is undefined at $x = 0$. When $x<0$, $y^\prime<0$, so the function is decreasing on the interval $(-\infty,0)$. When $x>0$, $y^\prime>0$, so the function is increasing on the interval $(0,\infty)$.

Step3: Analyze the shape of the graph.

Since it is an even function, it is symmetric about the y - axis. As $x\to\pm\infty$, $y = x^{\frac{6}{11}}\to+\infty$.

The correct graph is the one that is decreasing on $(-\infty,0)$ and increasing on $(0,\infty)$ and is symmetric about the y - axis.

Answer:

D. The function is decreasing on the interval $(-\infty,0)$ and increasing on the interval $(0,\infty)$.