The value c of the Rolle's theorem for the fu | vedclass

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

Question

(D) Given function $f(x) = 2 \sin x + \sin 2x$ on $[0, \pi]$.Since $f(x)$ is a sum of trigonometric functions,it is continuous on $[0, \pi]$ and diffe

Vedclass · Accepted Answer

$\frac{\pi}{3}$

Vedclass · Answer

(D) Given function $f(x) = 2 \sin x + \sin 2x$ on $[0, \pi]$.Since $f(x)$ is a sum of trigonometric functions,it is continuous on $[0, \pi]$ and differentiable on $(0, \pi)$.Also,$f(0) = 2 \sin(0) + \sin(0) = 0$ and $f(\pi) = 2 \sin(\pi) + \sin(2\pi) = 0$.Thus,$f(0) = f(\pi)$,satisfying the conditions of Rolle's theorem.We find $f'(x) = 2 \cos x + 2 \cos 2x$.Setting $f'(c) = 0$,we get $2 \cos c + 2 \cos 2c = 0$,which implies $\cos c + \cos 2c = 0$.Using the identity $\cos 2c = 2 \cos^2 c - 1$,we have $2 \cos^2 c + \cos c - 1 = 0$.Factoring the quadratic equation: $(2 \cos c - 1)(\cos c + 1) = 0$.This gives $\cos c = \frac{1}{2}$ or $\cos c = -1$.For $c \in (0, \pi)$,$\cos c = \frac{1}{2}$ implies $c = \frac{\pi}{3}$.Since $\cos c = -1$ gives $c = \pi$,which is not in the open interval $(0, \pi)$,the only valid value is $c = \frac{\pi}{3}$.

The value $c$ of the Rolle's theorem for the function $f(x) = 2 \sin x + \sin 2x$ in the interval $[0, \pi]$ is

Similar Questions

If $a + b + c = 0$,then how many roots does the equation $3ax^2 + 2bx + c = 0$ have in the interval $(0, 1)$?

The value of $c$ in the Lagrange's mean value theorem for $f(x)=\sqrt{x-2}$ in the interval $[2,6]$ is

If the function $f(x) = 2x^2 + 3x + 5$ satisfies the Lagrange's Mean Value Theorem $(LMVT)$ at $x = 3$ on the closed interval $[1, a]$,then the value of $a$ is equal to:

If Rolle's theorem holds for the function $f(x) = 2x^3 + ax^2 + bx$ in the interval $[-1, 1]$ for the point $c = \frac{1}{2}$,then the value of $2a + b$ is

If $f(x)=(2x-1)(3x+2)(4x-3)$ is a real-valued function defined on $[\frac{1}{2}, \frac{3}{4}]$, then the value$(s)$ of '$c$' as defined in the statement of Rolle's theorem is/are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module