Question

Given function f(x) = x^3 + 2x, find value of x = c that satisfies Mean-Value Theorem on the interval [- 1,1].

Answer 1

The solution is:

`\begin{equation*} x=\pm(√(3))/(3) \end{equation*}`

Explanation:

First, we'll calculate the function's average rate of change over [-1,1] as following:

`\begin{gathered} (f(-1)-f(1))/(-1-1)=-(f(-1)-f(1))/(2)=-(-3-3)/(2)=(6)/(2)=3 \\ \end{gathered}`

Now, we make f'(x) = 3 and solve for x, as following:

`\begin{gathered} f^(\prime)(x)=3x{}^2+2=3 \\ \\ \rightarrow3x^2=1\rightarrow x^2=(1)/(3)\rightarrow x=\pm(1)/(√(3)) \\ \\ \Rightarrow x=\pm(√(3))/(3) \\ \end{gathered}`

Therefore, we can conlcude that the solution is:

`\begin{equation*} x=\pm(√(3))/(3) \end{equation*}`