If f(x)=sin x+2 cos ^{2} x for frac{pi}{4} le | vedclass

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

Question

(C) Given, $f(x)=\sin x+2 \cos ^{2} x$ for $x \in \left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$.$f'(x) = \cos x - 4 \cos x \sin x = \cos x(1 - 2 \sin 2

Vedclass · Accepted Answer

minimum at $x=\frac{\pi}{2}$

Vedclass · Answer

(C) Given, $f(x)=\sin x+2 \cos ^{2} x$ for $x \in \left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$.$f'(x) = \cos x - 4 \cos x \sin x = \cos x(1 - 2 \sin 2x)$ is incorrect. Let us differentiate correctly:$f'(x) = \cos x + 2(2 \cos x)(-\sin x) = \cos x - 2 \sin 2x$.Setting $f'(x) = 0$, we have $\cos x(1 - 4 \sin x) = 0$.This gives $\cos x = 0$ or $\sin x = \frac{1}{4}$.For $x \in \left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$, $\cos x = 0 \Rightarrow x = \frac{\pi}{2}$.Also, $\sin x = \frac{1}{4}$ has no solution in this interval since $\sin x \geq \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \approx 0.707 > 0.25$.Checking the critical point and endpoints:$f\left(\frac{\pi}{4}\right) = \sin \frac{\pi}{4} + 2 \cos^2 \frac{\pi}{4} = \frac{1}{\sqrt{2}} + 2(\frac{1}{2}) = \frac{1}{\sqrt{2}} + 1 \approx 1.707$.$f\left(\frac{\pi}{2}\right) = \sin \frac{\pi}{2} + 2 \cos^2 \frac{\pi}{2} = 1 + 0 = 1$.$f\left(\frac{3 \pi}{4}\right) = \sin \frac{3 \pi}{4} + 2 \cos^2 \frac{3 \pi}{4} = \frac{1}{\sqrt{2}} + 2(\frac{1}{2}) = \frac{1}{\sqrt{2}} + 1 \approx 1.707$.Thus, $f(x)$ attains its minimum at $x = \frac{\pi}{2}$.

If $f(x)=\sin x+2 \cos ^{2} x$ for $\frac{\pi}{4} \leq x \leq \frac{3 \pi}{4}$, then $f$ attains its

Similar Questions

If $m$ and $M$ are the absolute minimum and absolute maximum values of the function $f(x) = 2\sqrt{2} \sin x - \tan x$ in the interval $[0, \pi/3]$,then $m + M =$

The perimeter of a sector is $p$. The area of the sector is maximum when its radius is

If $x = -1$ and $x = 2$ are the extreme points of the function $y = a \log|x| + bx^2 + x$,then find the values of $a$ and $b$.

The sum of two non-zero numbers is $4$. The minimum value of the sum of their reciprocals is

If the function $f(x)=2x^{3}-9ax^{2}+12a^{2}x+1$ where $a>0$ attains its maximum and minimum at $p$ and $q$ respectively such that $p^{2}=q$, then $a$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module