Verify Rolle's theorem of the following function on the indicated… - Vidyadip

Question

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\sin3\text{x}\text{ on }[0,\pi]$

✓

Answer

The given function is $\text{f}(\text{x})=\sin3\text{x}$
Since $\sin3\text{x}$ everywhere continuous and differentiable,
$\sin3\text{x}$ 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
$\text{f}(\text{x})=\sin3\text{x}$
$\Rightarrow \text{f}'(\text{x})=3\cos3\text{x}$
$\therefore\ \text{f}'(\text{x})=0$
$\Rightarrow3\cos3\text{x}=0$
$\Rightarrow\cos3\text{x}=0$
$\Rightarrow3\text{x}=\frac{\pi}{2},\frac{3\pi}{2},....$
$\Rightarrow\text{x}=\frac{\pi}{6},\frac{\pi}{2},\frac{5\pi}{6}$
Since, $\text{c}=\frac{\pi}{4}\in(0,\pi)$ 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

Using differentials, find the approximate values of the following:
$(66)^{\frac{1}{3}}$

If $\vec{\text{a}}=\text{a}_1\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}},\vec{\text{b}}=\text{b}_1\hat{\text{i}}+\text{b}_2\hat{\text{j}}+\text{b}_3\hat{\text{k}}$ and $\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}},$ then verify that $\vec{\text{a}}\times\big(\vec{\text{b}}+\vec{\text{c}}\big)=\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{a}}\times\vec{\text{c}}.$

Using vector method, prove that the point is collinear:
A(-3, -2, -5), B(1, 2, 3) and C(3, 4, 7)

Evaluate the following integrals:
$\int\text{x}\sqrt{\text{x}^4+1}\text{dx}$

Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{ae}^{\theta}(\sin\theta-\cos\theta),\text{y}=\text{ae}^\theta(\sin\theta+\cos\theta)$

By examining the chest $X-$ray, probability that $T.B.$ is detected when a person is actually suffering is $0.99.$ The probability that the doctor diagnoses incorrectly that a person has $T.B.$ on the basic of $X-$ray is $0.001.$ In a certain city $1$ in $1000$ persons suffers from $T.B.$ A person is selected at random is diagnosed to have $T.B$. what is the chance that he actually has $T.B.$?

The contents of three urns are as follows:

Urn $1 : 7$ white, $3$ black balls,

Urn $2 : 4$ white, $6$ black balls,

Urn $3 : 2$ white, $8$ black balls.

One of these urns is chosen at random with probabilities $0.20, 0.60$ and $0.20$ respectively. From the chosen urn two balls are drawn at random without replacement. If both these balls are white, what is the probability that these came from urn $3$?

Determine whether the following pair of lines intersect or not:
$\frac{\text{x}-5}{4}=\frac{\text{y}-7}{4}=\frac{\text{z}+3}{-5}$ and $\frac{\text{x}-8}{7}=\frac{\text{y}-4}{1}=\frac{3-5}{3}$

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{3}}_{-\frac{\pi}{3}}\frac{1}{1+\text{e}^{\tan\text{x}}}\text{ dx}$

If the marginal cost of maufacturing a certain item is given by $\text{C}(\text{x})=\frac{\text{dC}}{\text{dx}}=2+0.15\text{x}$. Find the total cost function C(x), given that C(0) = 100.