Download App

Mean Value Theorems — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsMean Value Theorems5 Marks
Question
If the value of c prescribed in Roll's theorem for the function$\text{f}(\text{x})=2\text{x}(\text{x}-3)^{\text{n}}$ on the interval $\big[0,2\sqrt3\big]$ is $\frac{3}{4},$ write the value of n (a positive integers).
✓

Answer

We have,
$\text{f}(\text{x})=2\text{x}(\text{x}-3)^{\text{n}}$
Differentiating the given function with respect to x, we get
$\text{f}'(\text{x})=2\big[\text{xn}(\text{x}-3)^{\text{n}-1}+(\text{x}-3)^{\text{n}}\big]$
$\Rightarrow\text{f}'(\text{x})=2(\text{x}-3)^{\text{n}}\Big[\frac{\text{xn}}{(\text{x}-3)}+1\Big]$
$\Rightarrow\text{f}'(\text{c})=2(\text{c}-3)^{\text{n}}\Big[\frac{\text{cn}}{(\text{c}-3)}+1\Big]$
Given:
$\text{f}'\Big(\frac{3}{4}\Big)=0$
$\therefore\ 2-\Big(\frac{9}{4}\Big)^{\text{n}}\Bigg[\frac{\frac{3}{4}\text{n}}{\big(\frac{-9}{4}\big)}+1\Bigg]=0$
$\Rightarrow2-\Big(\frac{9}{4}\Big)^{\text{n}}\Big[\frac{-\text{n}}{3}+1\Big]=0$
$\Rightarrow\Big[\frac{-\text{n}}{3}+1\Big]=0$
$\Rightarrow-\text{n}+3=0$
$\Rightarrow\text{n}=3$

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 "Solve the Following Question.(5 Marks)" from Mean Value Theorems→Mean Value Theorems — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5kg of flour and 1kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.
→
Given the probability that A can solve a problem is $\frac{2}{3}$ and the probability that B can solve the same problem is $\frac{3}{5}$. Find the probability that none of the two will be able to solve the problem.
→
Find the local maxima or local minima of $f(x)=x^3-3 x$.
→
Find the equation of the plane passing through the line of intersection of the planes $2x - y = 0$ and $3z - y= 0$ and perpendicular to the plane $4x + 5y - 3z = 8$.
→
$\int(2\text{x}^2+3)\sqrt{\text{x+2}}\text{dx}$
→
Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 2 & 3 & 1 \\ 2 & 4 & 1 \\ 3 & 7 & 2 \end{bmatrix}$
→
Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹ x each, ₹ y each and ₹ z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹ 1,000. School Q wants to spend ₹ 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹ 600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.
→
Evaluate :

$\int \sqrt{\frac{x-7}{x-9}} d x$

→
Let X denote the sum of numbers obtained when two fair dice are rolled. Find the standard deviation of X.
→
If $\text{P}=\begin{bmatrix}\text{x}&0&0\\0&\text{y}&0\\0&0&\text{z}\end{bmatrix}$ and $\text{Q}=\begin{bmatrix}\text{a}&0&0\\0&\text{b}&0\\0&0&\text{c}\end{bmatrix},$ prove that $\text{PQ}=\begin{bmatrix}\text{xa}&0&0\\0&\text{y}\text{b}&0\\0&0&\text{zc}\end{bmatrix}=\text{QP}$
→