Verify mean value theorem for the function: f(x)=1 4x-1 in [1,4]. - Vidyadip

Question

Verify mean value theorem for the function:
$\text{f(x)}=\frac{1}{4\text{x}-1}\text{ in }[1,4].$

✓

Answer

We have, $\text{f(x)}=\frac{1}{4\text{x}-1}\text{ in }[1,4]$
f(x) is continuous in [1, 4].
Also, at $\text{x}=\frac{1}{4}.$ f(x) is discontinuous.
Hence, f(x) is continuous in [1, 4].
$\text{f}'(\text{x})=-\frac{4}{(4\text{x}-1)^2},$ which exists in (1, 4).
Since, conditions of mean value theorem are satisfied.
Hence, there exists a real number $\text{c}\in[1,4]$ such that
$\text{f}'(\text{c})=\frac{\text{f}(4)-\text{f}(1)}{4-1}$
$\Rightarrow\ \frac{-4}{(4\text{c}-1)^2}=\frac{\frac{1}{16-1}-\frac{1}{4-1}}{4-1}=\frac{\frac{1}{15}-\frac{1}{3}}{3}$
$\Rightarrow\ \frac{-4}{(4\text{c}-1)^2}=\frac{1-5}{45}=\frac{-4}{45}$
$\Rightarrow\ (4\text{c}-1)^2=45$
$\Rightarrow\ 4\text{c}-1=\pm3\sqrt{5}$
$\Rightarrow\ \text{c}=\frac{3\sqrt{5}+1}{4}\in(1,4)$ [neglecting (-ve) value]
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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also find mean and variance of the distribution.

If $\text{A}=\frac{1}{9}\begin{bmatrix}-8 & 1 & 4\\4 & 4 & 7 \\ 1 & -8 & 4 \end{bmatrix},$ prove that $A^{-1} = A^3.$

For the following pairs of matrices verify that $(AB)^{-1} = B^{-1} A^{-1}$:
$\text{A}=\begin{bmatrix}3 & 2 \\7 & 5 \end{bmatrix}\text{ and B}=\begin{bmatrix}4 & 6 \\3 & 2 \end{bmatrix}$

Differentiate the following functions with respect to x:
$\sin^{-1}\Big\{\frac{\sin\text{x}+\cos\text{x}}{\sqrt{2}}\Big\},-\frac{3\pi}{4}<\text{x}<\frac{\pi}{4}$

Solve the following differential equation:$\frac{\text{dy}}{\text{dx}}+2\text{y}=6\text{e}^{\text{x}}$

If $\cos\text{y}=\text{x}\cos(\text{a}+\text{y}),$ where $\cos\text{a}\neq\pm1,$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\cos^2(\text{a}+\text{y})}{\sin\text{a}}$

Mother, father and son line up at random for a family picture. If A and B are two events given by A = Son on one end, B = Father in the middle, find P (A/B) and P (B/A).

Solve the following differential equations:$\cos\text{x}\cos\text{y}\frac{\text{dy}}{\text{dx}}=-\sin\text{x}\sin\text{y}$

Evaluate the following integrals:
$\int\sin^3\sqrt{\text{x}}\text{dx}$

Evaluate the following intregals:
$\int\frac{1}{\text{x}(\text{x}^4+1)}\ \text{dx}$