Download App

Mean Value Theorems — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMean Value Theorems5 Marks
Question
Verify the hypothesis and conclusion of Lagrange's mean value theorem for the function
$\text{f}(\text{x})=\frac{1}{4\text{x}-1},1\leq\text{x}\leq4.$
✓

Answer

The given function $\text{f}(\text{x})=\frac{1}{4\text{x}-1}.$ Clearly, f(x) is does not exist for x = 0 Since for each $\text{x}\in[1,4],$ the function attains a unique definite value, f(x) is continuous on [1, 4].Also, $\text{f}'(\text{x})=\frac{-4}{(4\text{x}-1)^2}$ exists for all $\text{x}\in[1,4],$
Thus, both the conditions of Lagrange's mean value theorem are verified. Concequently, there exists some $\text{c}\in[1,4]$ such that$\text{f}'(\text{c})=\frac{\text{f}(4)-\text{f}(1)}{4-1}$
$=\frac{\text{f}(4)-\text{f}(1)}{3}$ Now, $\text{f}(\text{x})=\frac{1}{4\text{x}-1}\Rightarrow\text{f}'(\text{x})=\frac{-4}{(4\text{x}-1)^2}$ $\text{f}(4)=\frac{1}{15},\text{f}(1)=\frac{1}{3}$ $\therefore\ \text{f}'(\text{x})=\frac{\text{f}(4)-\text{f}(1)}{4-1}$ $\Rightarrow\text{f}'(\text{x})=\frac{\frac{1}{15}-\frac{1}{3}}{4-1}==\frac{-4}{45}$ $\Rightarrow\frac{-4}{(4\text{x}-1)^2}=\frac{-4}{45}$ $\Rightarrow(4\text{x}-1)^2=45$ $\Rightarrow16\text{x}^2-8\text{x}-44=0$ $\Rightarrow4\text{x}^2-2\text{x}-11=0$ $\Rightarrow\text{x}=\frac{1}{4}\big(1+3\sqrt{5}\big)$ Thus, $\text{c}=\frac{1}{4}\big(1+3\sqrt{5}\big)\in(1,4)$ such that $\text{f}'(\text{c})=\frac{\text{f}(4)-\text{f}(1)}{4-1}.$ Hence, Lagrange'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 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

Maximum Z = 3x + 4y Subject to$\text{x}+\text{y}\leq30000$
$\text{y}\leq12000$
$\text{x}\geq6000$
$\text{x}\geq\text{y}$
$\text{x},\text{y}\geq0$
→
In the following, determine the values of constants involved in the definition so that the given function is continuous:
$\text{f(x)}=\begin{cases}4,&\text{if }\text{ x}\leq-1\\\text{ax}^2+\text{b},&\text{if }-1<\text{ x}<0\\\cos\text{x},&\text{if }\text{ x}\geq0\end{cases}$
→
Evaluate the following intregals:
$\int\frac{1}{\sin\text{x}-\sqrt{3}\cos\text{x}}\ \text{dx}$
→
If $\text{x}-\text{e}^{\tan\text{x}}+\sqrt{\frac{\text{x}^2+1}{2}},$ find $\frac{\text{dy}}{\text{dx}}$
→
Find:
$\int \frac{\cos \theta}{(4 + \sin^{2} \theta) (5 - 4 \cos^{2} \theta)} \text{d} \theta$
→
Show that the points A, B, C with position vectors $\vec{\text{a}}-2\vec{\text{b}}+3\vec{\text{c}},\ 2\vec{\text{a}}+3\vec{\text{b}}-4\vec{\text{c}}$ and $-7\vec{\text{b}}+10\vec{\text{c}}$ are collinear.
→
Solve the following equation for x:
$\tan^{-1}\Big(\frac{\text{x}-2}{\text{x}-4}\Big)+\tan^{-1}\Big(\frac{\text{x}+2}{\text{x}+4}\Big)=\frac{\pi}{4}$
→
Show that the lines
$\vec{\text{r}}=3\hat{\text{i}}+2\hat{\text{j}}-4\hat{\text{k}}+\lambda\big(\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}\big)$ and $\vec{\text{r}}=5\hat{\text{i}}-2\hat{\text{j}}+\mu\big(3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}\big)$ are intersecting. Hence, find their point of intersection.
→
A firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost of engaging each large van is Rs 400 and each small van is Rs 200. Not more than Rs 3000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost.
→
Evaluate the following intregals:
$\int\frac{\text{x}}{\sqrt{\text{x}^2+6\text{x}+10}}\ \text{dx}$
→