Discuss the applicability of the Rolle's theorem for the followin… - Vidyadip

Question

Discuss the applicability of the Rolle's theorem for the following function on the indicated interval
$\text{f}(\text{x})=[\text{x}]\text{ for }-1\leq\text{x}\leq1,$ where [x] denotes the greatest integer not exceeding x.

✓

Answer

Here, $\text{f}(\text{x})=[\text{x}]$ and $\text{x}\in[-1,1],$ at n = 1 $\text{LHL}=\lim\limits_{\text{x}\rightarrow(1-\text{h})}[\text{x}]$ $=\lim\limits_{\text{h}\rightarrow0}[1-\text{h}]$ $=0$ $\text{RHL}=\lim\limits_{\text{x}\rightarrow(1+\text{h})}[\text{x}]$ $=\lim\limits_{\text{h}\rightarrow0}[1+\text{h}]$ $=1$ $\text{LHL}\neq\text{RHL}$So, f(x) is not continuos at $1\in[-1,1]$
Hence, Rolle's theorem is not applicable on f(x) in [-1, 1].

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 "3 Marks Question" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Examine the continuity of the function $f(x) = x^3 + 2x^2 - 1$ at $x = 1.$

Solve the following equation for x:
$\tan^{-1}(\text{x}+1)+\tan^{-1}(\text{x}-1)=\tan^{-1}\frac{8}{31}$

ABCDE is a pentagon, prove that,$\overrightarrow{\text{AB}}+\overrightarrow{\text{AE}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{DC}}+\overrightarrow{\text{ED}}+\overrightarrow{\text{AC}}=3\ \overrightarrow{\text{AC}}$

If $\text{A}=\begin{bmatrix}1&2\\0&3 \end{bmatrix}$ is written as B + C, where B is a symmetric matrix and C is a skew- symmetric matrix, then B is equal to.

Find the area of a parallelogram whose adjacent sides are given by the vectors $\vec{a}=3 \hat{i}+\hat{j}+4 \hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$

Write the value of $\cos^{-1}\Big(\tan\frac{3\pi}{4}\Big).$

Prove that the function $f : R \rightarrow R$ defined by $f (x) = 2x + 5$ is one$-$one.

Manufacturer can sell x items at a price of rupees $(5-\frac{\text{x}}{100})$ each. The cost price is Rs $\Big(\frac{\text{x}}{5}+500\Big)$. Find the number of items he should sell to earn maximum profit.

For what value of k is the function
$\text{f}\text{(x)}=\begin{cases}\frac{\sin2\text{x}}{\text{x}}, & \text{x} \neq 0\\\text{k}, &\text{x} = 0\end{cases}$ continuous at x = 0.

Write the angle between the curves $y = e^{-x}$ and $y = e^x$ at their point of intersections.