If the function f(x)=sqrt{x^2-4} satisfies th | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The function $f(x) = \sqrt{x^2-4}$ is continuous on $[2, 4]$ and differentiable on $(2, 4)$.According to Lagrange's Mean Value Theorem,there exist

Vedclass · Accepted Answer

$\sqrt{6}$

Vedclass · Answer

(C) The function $f(x) = \sqrt{x^2-4}$ is continuous on $[2, 4]$ and differentiable on $(2, 4)$.According to Lagrange's Mean Value Theorem,there exists at least one $C \in (2, 4)$ such that $f'(C) = \frac{f(4) - f(2)}{4 - 2}$.First,calculate the derivative: $f'(x) = \frac{1}{2\sqrt{x^2-4}} 	imes 2x = \frac{x}{\sqrt{x^2-4}}$.Next,calculate the values at the endpoints: $f(4) = \sqrt{16-4} = \sqrt{12} = 2\sqrt{3}$ and $f(2) = \sqrt{4-4} = 0$.Substitute these into the formula: $\frac{C}{\sqrt{C^2-4}} = \frac{2\sqrt{3} - 0}{4 - 2} = \frac{2\sqrt{3}}{2} = \sqrt{3}$.Squaring both sides: $\frac{C^2}{C^2-4} = 3$.$C^2 = 3(C^2 - 4) \Rightarrow C^2 = 3C^2 - 12 \Rightarrow 2C^2 = 12 \Rightarrow C^2 = 6$.Since $C \in (2, 4)$,we take the positive root: $C = \sqrt{6}$.

If the function $f(x)=\sqrt{x^2-4}$ satisfies the Lagrange's mean value theorem on $[2, 4]$,then the value of $C$ is

Similar Questions

If the function $f(x) = x(x + 3) e^{-x/2}$ satisfies Rolle's theorem in the interval $[-3, 0]$,then find the value of $c$.

Let $f$ and $g$ be twice differentiable even functions on $(-2, 2)$ such that $f(\frac{1}{4}) = 0, f(\frac{1}{2}) = 0, f(1) = 1$ and $g(\frac{3}{4}) = 0, g(1) = 2$. Then,the minimum number of solutions of $f(x)g''(x) + f'(x)g'(x) = 0$ in $(-2, 2)$ is equal to

Which of the following functions can satisfy Rolle's theorem on the given interval?

If the function $f(x) = ax^3 + bx^2 + 11x - 6$, defined on $[1, 3]$, satisfies all the conditions of Rolle's theorem for $c = 2 + \frac{1}{\sqrt{3}}$, then

If $2a + 3b + 6c = 0$,then at least one root of the equation $ax^2 + bx + c = 0$ lies in the interval:

Vedclass Test Series

Exam Paper Generator

Online Exam Module