Download App

Mean Value Theorems — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMean Value Theorems5 Marks
Question
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point $'c\ '$ in the indicated interval as stated by the Lagrange's mean value theorem. $f(x) = 2x - x^{2 }$ on $[0, 1]$
✓

Answer

We have,f$(x) = 2x - x^{2 }$
Since a polynomial function is everywhere continuous and differentiable.
Therefore, f$(x)$ is continuous on $0, 1$ and differentiable on $0,1$
Thus, both conditions of Lagrange's mean value theorem are satisfied.
So, there must exist at least one real number $\text{c}\in0,1$ such that
$\text{f}'(\text{c})=\frac{\text{f}(1)-\text{f}(0)}{1-0}=\frac{\text{f}(1)-\text{f}(0)}{1}$
Now,
$f(x) = 2x - x^{2 }$
$\Rightarrow f'(x) = 2 - 2x,$
$\Rightarrow f(1) = 2 - 1$
$\Rightarrow f(1) = 1,$
$\Rightarrow f(0) = 0$
$\therefore\ \text{f}'(\text{x})=\frac{\text{f}(1)-\text{f}(0)}{1-0}$
$\Rightarrow2-2\text{x}=\frac{1-0}{1}$
$\Rightarrow-2\text{x}=1-2$
$\Rightarrow\text{x}=\frac{1}{2}$
Thus, $\text{c}=\frac{1}{2}\in(1,0)$ such that $\text{f}'(\text{c})=\frac{\text{f}(1)-\text{f}(0)}{1-0}$
Hence, Lagrange's mean value 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 Mean Value Theorems→Mean Value Theorems — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following integrals:$\int\text{e}^{\text{x}}\frac{1+\text{x}}{(2+\text{x})^2}\text{dx}$
→
By using the properties of definite integrals, evaluate the integral $\int_{0}^{\pi} \log (1+\cos x) d x$
→
An unbiased coin is tossed 4 times, Find the mean and variance of the number of heads obtained.
→
Discuss the continuity of the following functions:
  1. $\text{f(x)}=\sin\text{x}+\cos\text{x}$
  2. $\text{f(x)}=\sin\text{x}-\cos\text{x}$
  3. $\text{f(x)}=\sin\text{x}\cos\text{x}$
→
Solve the following initial value problems:
$(1+\text{y}^2)\text{dx}+(\text{x}-\text{e}^{\tan^{-1}\text{y}})\text{dy}=0,\text{ y}(0)=0$
→
An urn contains $m$ white and $n$ black balls. A ball is drawn at random and is put back into the urn along with $k$ additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. Show that the probability of drawing a white ball now does not depend on $k.$
→
Prove that $2 \tan ^{-1}\left\{\sqrt{\frac{\alpha-\beta}{\alpha+\beta}} \tan \frac{x}{2}\right\}=\cos ^{-1}\left(\frac{\beta+\alpha \cos x}{\alpha+\beta \cos x}\right)$
→
Represent the following families of curves by forming the corresponding differential equation:
$\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1$
→
Evaluate the following integrals:$\int\text{cosec}^3\text{x dx}$
→
Find the value of $4\tan^{-1}\frac{1}{5}-\tan^{-1}\frac{1}{239}.$
→