Question
Verify mean value theorem for the function:
$\text{f(x)}=\sin\text{x}-\sin2\text{x in }[0,\pi].$
$\text{f(x)}=\sin\text{x}-\sin2\text{x in }[0,\pi].$
✓
Answer
We have, $\text{f(x)}=\sin\text{x}-\sin2\text{x in }[0,\pi]$
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.
- Since, we know that sine functions are continuous functions hence
- $\text{f}'(\text{x})=\cos\text{x}-\cos2\text{x}.2=\cos\text{x}-2\cos2\text{x},$ which exists in $(0,\pi)$
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.
Explore more
Similar questions
Using matrices, solve the following system of equations:$x + 2y - 3z = 6$
$3x + 2y - 2z =3$
$2x - y + z = 2$
→$3x + 2y - 2z =3$
$2x - y + z = 2$
Find the particular solution of the differential equation satisfying the given conditions:
$\frac{\text{dy}}{\text{dx}}$= y tan x, given that y = 1 when x = 0.
→$\frac{\text{dy}}{\text{dx}}$= y tan x, given that y = 1 when x = 0.
Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}2&-1&0&1\\-3&0&1&-2\\1&1&-1&1\\2&-1&5&0 \end{vmatrix}$
→$\text{A}=\begin{vmatrix}2&-1&0&1\\-3&0&1&-2\\1&1&-1&1\\2&-1&5&0 \end{vmatrix}$
Two groups are competing for the positions of the Board of Directors of a Corporation. The probabilities that the first and the second groups will win are $0.6$ and $0.4$ respectively. Further, if the first group wins, the probability of introducing a new product is $0.7$ and the corresponding probability is $0.3$ if the second group wins. Find the probability that the new product introduced was by the second group.
→The feasible region for a LPP is shown in Evaluate Z = 4x + y at each of the corner points of this region. Find the minimum value of Z, if it exists.
A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:
The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP.
|
|
Product A
|
Product B
|
Weekly capacity
|
|
Department 1
|
3
|
2
|
130
|
|
Department 2
|
4
|
6
|
260
|
|
Selling price per unit
|
Rs. 25
|
Rs. 30
|
|
|
Labour cost per unit
|
Rs. 16
|
Rs. 20
|
|
|
Raw material cost per unit
|
Rs. 4
|
Rs. 4
|
|
A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?
A = the number of heads is two,
B = the last throw results in head.
A = the number of heads is two,
B = the last throw results in head.
If $\text{f(x)}=\begin{cases}\frac{\cos^2\text{x}-\sin^2\text{x}}{\sqrt{\text{x}^2+1}-1},&\text{x}\neq0\\\text{k},&\text{x}=0\end{cases}$ is continuous at x = 0, find k.
→ABCD is aparallelogram. the position vectora of the points A, B and C are respectively, $4\hat{\text{i}}+5\hat{\text{j}}-10\hat{\text{k}},2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$ and $-\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}.$ Find the vector equation of the line BD. Also, reduce it to cartesian form.
→Evaluate the following integrals:
$\int\frac{(\text{x}\tan^{-1}\text{x})}{(1+\text{x}^2)^{\frac{3}{2}}}\text{dx}$
→$\int\frac{(\text{x}\tan^{-1}\text{x})}{(1+\text{x}^2)^{\frac{3}{2}}}\text{dx}$