Question

If $f\left(x\right)=x^3$ and $g\left(x\right)=x^3-4x$ in $-2\le x\le 2$, then consider the statements

I $f\left(x\right)$ and $g\left(x\right)$ satisfy mean value theorem.

II $f\left(x\right)$ and $g\left(x\right)$ both satisfy Rolle's theorem.

Which of these statements is true?

• A. I and II are correct
• B. Only I is correct
• C. None is correct
• D. Only II is correct

Answer 1

The correct option is B

Only I is correct

Explanation for the correct option.

Step: 1- Concept used:

A function $f\left(x\right)$ satisfy mean value theorem when it is continuous and differentiable in the interval $\left[a,b\right]$.

A function $f\left(x\right)$ satisfy Rolle's theorem when it is continuous and differentiable in the interval $\left[a,b\right]$ and also $f\left(a\right)=f\left(b\right)$.

Step 2. Check the function $f\left(x\right)$.

The function $f\left(x\right)=x^3$ is a polynomial function so it is continuous and differentiable in the interval $\left[-2,2\right]$ and so it satisfies the mean value theorem.

For $x=2$,

$\begin{array}{rcl}f\left(2\right)& =& 2^3\\ & =& 8\end{array}$

and for $x=-2$

$\begin{array}{rcl}f\left(-2\right)& =& {\left(-2\right)}^3\\ & =& -8\end{array}$

Now as $f\left(2\right)\ne f\left(-2\right)$, so the function $f\left(x\right)$ does not satisfy the Rolle's theorem in the interval $\left[-2,2\right]$.

Step 3. Check the function $g\left(x\right)$.

The function $g\left(x\right)=x^3-4x$ is a polynomial function so it is continuous and differentiable in the interval $\left[-2,2\right]$ and so it satisfies the mean value theorem.

For $x=2$,

$\begin{array}{rcl}g\left(2\right)& =& 2^3-4\times 2\\ & =& 8-8\\ & =& 0\end{array}$

and for $x=-2$

$\begin{array}{rcl}g\left(-2\right)& =& {\left(-2\right)}^3-4\left(-2\right)\\ & =& -8+8\\ & =& 0\end{array}$

Now as $g\left(2\right)=g\left(-2\right)$, so the function $g\left(x\right)$ satisfies the Rolle's theorem in the interval $\left[-2,2\right]$.

Thus both the functions $f\left(x\right)$ and $g\left(x\right)$ satisfies the mean value theorem but only $g\left(x\right)$ satisfies the Rolle's theorem.

So statement I is correct but statement II is incorrect.

Hence, the correct option is B.