Answer :
Answer:
The solution is:
[tex]\begin{equation*} x=\pm\frac{\sqrt{3}}{3} \end{equation*}[/tex]Step-by-step explanation:
First, we'll calculate the function's average rate of change over [-1,1] as following:
[tex]\begin{gathered} \frac{f(-1)-f(1)}{-1-1}=-\frac{f(-1)-f(1)}{2}=-\frac{-3-3}{2}=\frac{6}{2}=3 \\ \end{gathered}[/tex]Now, we make f'(x) = 3 and solve for x, as following:
[tex]\begin{gathered} f^{\prime}(x)=3x{}^2+2=3 \\ \\ \rightarrow3x^2=1\rightarrow x^2=\frac{1}{3}\rightarrow x=\pm\frac{1}{\sqrt{3}} \\ \\ \Rightarrow x=\pm\frac{\sqrt{3}}{3} \\ \end{gathered}[/tex]Therefore, we can conlcude that the solution is:
[tex]\begin{equation*} x=\pm\frac{\sqrt{3}}{3} \end{equation*}[/tex]