Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\frac{\sin\text{x}}{\text{e}^{\text{x}}}\text{ on }0\leq\text{x}\leq\pi$
✓

Answer

The given function is $\text{f}(\text{x})=\frac{\sin\text{x}}{\text{e}^{\text{x}}}.$
Since $\cos\text{x}$ and $\text{e}^\text{x}$ are everywhere continuous and differentiable, being a quotient of these two, f(x) is continuous on $[0,\pi]$ and differentiable on $(0,\pi).$
Also,
$\text{f}(\pi)=\text{f}(0)=0$
Thus, f(x) satisfies all the conditions of Rolle's theorem.
Now, we have to show that there exists $\text{c}\in(0,\pi)$ such that f'(c) = 0.
We have
$\text{f}(\text{x})=\frac{\sin\text{x}}{\text{e}^{\text{x}}}$
$\Rightarrow \text{f}'(\text{x})=\frac{\cos\text{x}-\sin\text{x}}{\text{e}^{\text{x}}}$
$\therefore\ \text{f}'(\text{x})=0$
$\Rightarrow\frac{\cos\text{x}-\sin\text{x}}{\text{e}^{\text{x}}}=0$
$\Rightarrow\cos\text{x}-\sin\text{x}=0$
$\Rightarrow\tan\text{x}=1$
$\Rightarrow\text{x}=\frac{\pi}{4}$
Thus, $\text{c}=\frac{\pi}{4}\in(0,\pi)$ such that f'(c) = 0
Hence, Rolle's 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 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

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}\sin^223^{\circ}&\sin^267^{\circ}&\cos180^{\circ}\\-\sin^267^{\circ}&-\sin^223^{\circ}&\cos^2180^{\circ}\\\cos180^{\circ}&\sin^223^{\circ}&\sin^267^{\circ}\end{vmatrix}$
→
If $\text{y}=500\text{e}^{7\text{x}}+600\text{e}^{-7\text{x}}$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=49\text{y}.$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{6}}_{0}\cos^{-3}2\theta\sin2\theta\text{ d}\theta$
→
Verify Rolle's theorem for the following function on the indicated intervals$\text{f}(\text{x})=\sin2\text{x}\text{ on }\Big[0,\frac{\pi}{2}\Big]$
→
Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}\frac{\sin3\text{x}}{\text{x}},&\text{if }\text{ x}\neq0\\4,&\text{if }\text{ x}=0\end{cases}$
→
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs.
  1. None
  2. Not more than one
  3. More than one
  4. At least one
will fuse after 150 days of use.
→
Solve the following differential equation:
$\big(\cot^{-1}\text{y} + \text{x}\big)\text{dy}= \big(1 + \text{y}^2\big) \text{dx}$
→
The cost of 4kg onion, 3kg wheat and 2kg rice is Rs. 60. The cost of 2kg onion, 4kg wheat and 6kg rice is Rs. 90. The cost of 6kg onion 2kg wheat and 3kg rice is Rs. 70. Find the cost of each item per kg by matrix method.
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}-\text{x}\text{e}^\text{x}-\frac{5}{2}+\cos^2\text{x}$
→
Form the differential equation of the family of hyperebolas having foci on x- axis and centre at the origine.
→