Discuss the applicability of Lagrange's mean value theorem for th… - Vidyadip

Question

Discuss the applicability of Lagrange's mean value theorem for the function:
f(x) = |x| on [−1, 1]

✓

Answer

Here,
f(x) = |x| on [−1, 1]
$\text{f}(\text{x})=\begin{cases}-\text{x},&\text{x}<0\\\text{x},&\text{x}\geq0\end{cases}$
For differentiability at x = 0
$\text{LHD}=\lim_\limits{\text{x}\rightarrow0^-}\frac{\text{f}(0-\text{h})-\text{f}(0)}{-\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{-(0-\text{h})-0}{-\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}}{-\text{h}}$
$\text{LHD}=-1$
$\text{RHD}=\lim_\limits{\text{x}\rightarrow0^+}\frac{\text{f}(0+\text{h})-\text{f}(0)}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{(0+\text{h})-0}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{h}}{\text{h}}$
$=1$
$\text{LHD}\neq\text{RHD}$
⇒ f(x) is not differentiable at $\text{x}=0\in(-1,1)$
Hence, Lagrange's mean value theorem is verified.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Mean Value Theorems→Mean Value Theorems — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\tan^{7}\text{x}}{\tan^{7}\text{x}+\cot^7\text{x}}\text{ dx}$

Maximize Z = 3x + 3y, if possible,
Subject to the constraints
$\text{x}-\text{y}\leq1$
$\text{x}+\text{y}\geq3$
$\text{x},\text{y}\geq0$

Evaluate the following intregals:
$\int\frac{5\text{x}}{(\text{x}+1)(\text{x}^2-4)}\text{dx}$

A bag contains $4$ white and $5$ black balls. Another bag contains $9$ white and $7$ black balls. $A$ ball is transferred from the first bag to the second and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, then what is the constitutional probability that both are girls? Given that.

The youngest is a girl,
At least one is a girl.

If $\text{A}=\begin{bmatrix}2&3\\1&2\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix},$ then find $\lambda,\mu$ so that $\text{A}^2=\lambda\text{A}+\mu\text{I}$

Solve the following differential equations:$\text{y}(1+\text{e}^{\text{x}})\text{dy}=(\text{y}+1)\text{e}^{\text{x}}\text{ dx}$

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\tan^{7}\text{x}}{\tan^{7}\text{x}+\cot^7\text{x}}\text{ dx}$

Evaluate the following integrals:
$\int\limits^2_1\frac{1}{\text{x}(1+\log\text{x})^2}\text{ dx}$

Evaluate the following:
$\begin{vmatrix}0&\text{xy}^2&\text{xz}^2\\\text{x}^2\text{y}&0&\text{yz}^2\\\text{x}^2\text{z}&\text{zy}^2&0\end{vmatrix}$