Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Verify Rolle's theorem for the following function on the indicated intervals$\text{f}(\text{x})=\cos2\Big(\text{x}-\frac{\pi}{4}\Big)\text{ on }\Big[0,\frac{\pi}{2}\Big]$
✓

Answer

The given function is $\text{f}(\text{x})=\cos2\Big(\text{x}-\frac{\pi}{4}\Big)$ $=\cos\Big(2\text{x}-\frac{\pi}{2}\Big)=\sin2\text{x}.$ Thus, we have to show that there exists $\text{c}\in\Big(0,\frac{\pi}{2}\Big)$ such that f'(c) = 0.We have
$\text{f}(\text{x})=\sin2\text{x}$ $\Rightarrow\text{f}'(\text{x})=2\cos2\text{x}$ $\Rightarrow\text{f}'(\text{x})=0$ $\Rightarrow2\cos2\text{x}=0$ $\Rightarrow\cos2\text{x}=0$ $\Rightarrow\text{x}=\frac{\pi}{4}$ Thus, $\text{c}=\frac{\pi}{4}\in\Big(0,\frac{\pi}{2}\Big)$ such that $\text{f}'(\text{c})=0.$ 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

Prove the following results:
$2\tan^{-1}\Big(\frac{1}{2}\Big)+\tan^{-1}\Big(\frac{1}{7}\Big)=\tan^{-1}\Big(\frac{31}{17}\Big)$
→
Evaluate the following intregals:
$\int\frac{3}{(1-\text{x})(1+\text{x}^2)}\ \text{dx}$
→
Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{x}=\text{a}(\theta-\sin\theta),\text{y}=\text{a}(1-\cos\theta)\ \text{at}\ \theta=\frac{-\pi}{2}$
→
If $\text{y}=\text{x}\sin^{-1}\text{x}+\sqrt{1-\text{x}^2},$ prove that $\frac{\text{dy}}{\text{dx}}=\sin^{-1}\text{x}$
→
Find the approximate change in the surface area of a cube of side $x$ metres caused by decreasing the side by $1\%.$
→
Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution.
→
If a unit vector $\vec a$ makes angles $\frac{\pi}{3}$ with $\hat i$, $\frac{\pi}{4}$ with $\hat j$ and an acute angle $\theta$ with $\hat k$, then find $\theta$ and hence, the components of $\vec a $.
→
Solve the following differential equation
$\text{xy}(\text{y}+1)\text{dy}=(\text{x}^2+1)\text{dx}$
→
Solve the following initial value problems $\tan\text{x}\Big(\frac{\text{dy}}{\text{dx}}\Big)=2\text{x}\tan\text{x}+\text{x}^2-\text{y},\tan\text{x}\neq0$ given that y = 0 when $\text{x}=\frac{\pi}{2}$
→
Find the derivative of sin (3x + 2) w.r.t. x from first principle.
→