Verify mean value theorem for the function: f(x)= - 2x in [0, ]. - Vidyadip

Question

Verify mean value theorem for the function:
$\text{f(x)}=\sin\text{x}-\sin2\text{x in }[0,\pi].$

✓

Answer

We have, $\text{f(x)}=\sin\text{x}-\sin2\text{x in }[0,\pi]$

Since, we know that sine functions are continuous functions hence

$\text{f(x)}=\sin\text{x}-\sin2\text{x}$ is a continuous function in $[0,\pi].$

$\text{f}'(\text{x})=\cos\text{x}-\cos2\text{x}.2=\cos\text{x}-2\cos2\text{x},$ which exists in $(0,\pi)$

So, f(x) is differentiable in $(0,\pi).$ Continuous of mean value theorem are satisfied.
Hence, $\exists\text{ c}\in(0,\pi)$ such that, $\text{f}'(\text{c})=\frac{\text{f}(\pi)-\text{f}(0)}{\pi-0}$
$\Rightarrow\ \cos\text{c}-2\cos2\text{c}=\frac{\sin\pi-\sin2\pi-\sin0+\sin2.0}{\pi-0}$
$\Rightarrow\ 2\cos2\text{c}-\cos\text{c}=\frac{0}{\pi}$
$\Rightarrow\ 2.(2\cos^2\text{c}-1)-\cos\text{c}=0$
$\Rightarrow\ 4\cos^2\text{c}-2-\cos\text{c}=0$
$\Rightarrow\ 4\cos^2\text{c}-\cos\text{c}-2=0$
$\Rightarrow\ \cos\text{c}=\frac{1\pm\sqrt{1+32}}{8}=\frac{1\pm\sqrt{33}}{8}$
$\therefore\ \text{c}=\cos^{-1}\Big(\frac{1\pm\sqrt{33}}{8}\Big)$
Also, $\cos^{-1}\Big(\frac{1\pm\sqrt{33}}{8}\Big)\in(0,\pi)$
Hence, mean value theorem has been 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

If $\text{A}=\begin{bmatrix}2 & 3 \\ 1 & 2 \end{bmatrix},$ verify that $A^2 - 4A + I = 0,$ where $\text{I}=\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}\text{ and O}\begin{bmatrix}0 & 0 \\ 0 & 0 \end{bmatrix}.$ Hence find $A^{-1}.$

Find non-zero values of x satisfying the matrix equation:$\text{x}\begin{bmatrix}2\text{x}&2\\3&\text{x}\end{bmatrix}+2\begin{bmatrix}8&5\text{x}\\4&4\text{x}\end{bmatrix}=2\begin{bmatrix}(\text{x}^2+8)&24\$10)&6\text{x}\end{bmatrix}$

Solve the following system of equations by matrix method:
$3x + 4y + 2z = 8$
$2y - 3z = 3$
$x - 2y + 6z = -2$

A manufacturer of Furniture makes two products : chairs and tables. processing of these products is done on two machines $A$ and $B$. A chair requires $2 hrs$ on machine $A$ and $6$ hrs on machine $B$. A table requires $4 hrs$ on machine $A$ and $2 hrs$ on machine $B.$ There are $16 hrs$ of time per day available on machine $A$ and $30 hrs$ on machine $B.$ Profit gained by the manufacturer from a chair and a table is $Rs. 3$ and $Rs. 5$ respectively. Find with the help of graph what should be the daily production of each of the two products so as to maximize his profit.

If $\text{y}=\text{e}^{2\text{x}}(\text{ax}+\text{b}),$ show that $\text{y}_2-\text{4}\text{y}_1+4\text{y}=0$

Solve the following system of equations by matrix method:
$3x + 7y = 4$
$x + 2y = -1$

A bag contains $(2n + 1)$ coins. It is known that $n$ of these coins have a head on both sides where as the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is $\frac{31}{42},$ determine the value of $n.$

Solve the following differential equation:
$\text{y}^2\frac{\text{dx}}{\text{dy}}+\text{x}-\frac{1}{\text{y}}=0$

How should we choose two numbers, each greater than or equal to −2, whose sum so that the sum of the first and the cube of the second is minimum?

Solve the following differential equation:
$(\text{x}+\text{y})^2\frac{\text{dy}}{\text{dx}} = 1$