Download App

Mean Value Theorems — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMean Value Theorems3 Marks
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 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

Find the shortest distance between the following lines: $\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k}) \ \vec{r}=(2 \hat{i}+4 \hat{j}+5 \hat{k})+\mu(4 \hat{i}+6 \hat{j}+8 \hat{k})$
→
Integrate the function in Exercise:
$\frac{\sin\text{x}}{\sin(\text{x}-\text{a)}}$
→
Find the integral value of x, if $\begin{vmatrix}\text{x}^2&\text{x}&1\\0&2&1\\3&1&4 \end{vmatrix}=28.$
→
Verify $A(\operatorname{adj} . A)=(\operatorname{adj} . A) A=|A| \mid$ : $\left[\begin{array}{cc} 2 & 3 \\ -4 & -6 \end{array}\right] $
→
Write the value of $\sin^{-1}\Big(\frac{-\sqrt3}{2}\Big)+\cos^{-1}\Big(\frac{-1}{2}\Big)$
→
Discuss the applicability of the Rolle's theorem for the following function on the indicated interval
$\text{f}(\text{x})=\begin{cases}-4\text{x}+5,&0\leq\text{x}\leq1\\2\text{x}-3,&1<\text{x}\leq2\end{cases}$
→
If $\text{f(x)}=\begin{cases}2\text{x}^2+\text{k},&\text{if }\text{ x}\geq0\\-2\text{x}^2+\text{k},&\text{if }\text{ x}<0\end{cases},$ then what should be the value of k so that f(x) is continuous at x = 0.
→
Compute the magnitude of the following vectors: $\vec a = \hat i + \hat j + \hat k,\vec b = 2\hat i - 7\hat j - 3\hat k,$ $\vec c = \frac{1}{{\sqrt 3 }}\hat i + \frac{1}{{\sqrt 3 }}\hat j - \frac{1}{{\sqrt 3 }}\hat k$
→
Reduce the equation 2x - 3y - 6z = 14 to the normal form and, hence, find the length of the perpendicular from the origin to the plane. Also, find the direction cosines of the normal to the plane.
→
Find the area of the triangle formed by O, A, B when $\overrightarrow{\text{OA}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}},\overrightarrow{\text{OB}}=-3\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}.$
→