Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\frac{6\text{x}}{\pi}-4\sin^{2}\text{x}\text{ on }\Big[0,\frac{\pi}{6}\Big]$
✓

Answer

The given function is $\text{f}(\text{x})=\frac{6\text{x}}{\pi}-4\sin^{2}\text{x}$ Since $\sin\text{x}\ \&\ \text{x}$ are everywhere continuous and differentiable, f(x) is continuous on $\Big[0,\frac{\pi}{6}\Big]$ and differentiable on $\Big(0,\frac{\pi}{6}\Big)$ Also, $\text{f}\Big(\frac{\pi}{6}\Big)=\text{f}(0)=0$ Thus, f(x) satisfies all the conditions of Rolle's theorem. Now, we have to show that there must exist a point $\text{c}\in\Big(0,\frac{\pi}{6}\Big)$ such that f'(c) = 0.We have
$\text{f}(\text{x})=\frac{6\text{x}}{\pi}-4\sin^{2}\text{x}$ $\Rightarrow\text{f}'(\text{x})=\frac{6}{\pi}-8\sin\text{x}\cos\text{x}$ $\therefore\ \text{f}'(\text{x})=0$ $\Rightarrow\frac{6}{\pi}-8\sin\text{x}\cos\text{x}=0$ $\Rightarrow\sin2\text{x}=\frac{3}{2\pi}$ $\Rightarrow\text{x}=\frac{1}{2}\sin^{-1}\Big(\frac{3}{2\pi}\Big)$ Thus, $\text{c}=\frac{1}{2}\sin^{-1}\Big(\frac{3}{2\pi}\Big)\in\Big(0,\frac{\pi}{6}\Big)$ 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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following intregals:
$\int\frac{5\text{x}}{(\text{x}+1)(\text{x}^2-4)}\text{dx}$
→
find the area of the region bounded by y = |x - 1| and y = 1.
→
Show that the following system of linear equations is consistent and also find solution:
$6x + 4y = 2$
$9x + 6y =3$
→
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.
→
In order to supplement daily diet, a person wishes to take some X and some wishes Y tablets. The contents of iron, calcium and vitamins in X and Y(in milligrams per tablet) are given as below:
Tablets
Iron
Calcium
Vitamin
x
6
3
2
y
2
3
4
The person needs at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligram of vitamins. The price of each tablet of X and Y is Rs. 2 and Rs. 1 respectively. How many tablets of each should the person take inorder to satisfy the above requirement at the minimum cost?
→
Discuss the continuity of the f(x) at the indicated points f(x) = |x| + |x - 1| at x = 0, 1.
→
If $\sin(\text{xy})+\frac{\text{y}}{\text{x}}=\text{x}^2-\text{y}^2,$ find $\frac{\text{dy}}{\text{dx}}$
→
Evaluate the following integrals:
$\int\limits^{\infty}_0\frac{\text{x}}{(1+\text{x})(1+\text{x}^2)}\text{ dx}$
→
If $\text{y}=(\sin\text{x})^{(\sin\text{x})^{(\sin\text{x})^{....\infty}}},$ prove that $\frac{\text{y}^2\cot\text{x}}{(1-\text{y}\log\sin\text{x})}$
→
Differentiate the following functions with respect to x:
$(\log\text{x})^{\cos\text{x}}$
→