The minimum value of 2^{sin x}+2^{cos x} is | vedclass

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

Question

(A) We know that by the $AM-GM$ inequality,for positive real numbers $a$ and $b$,$\frac{a+b}{2} \geq \sqrt{ab}$.Applying this to $2^{\sin x}$ and $2^{

Vedclass · Accepted Answer

$2^{1-1 / \sqrt{2}}$

Vedclass · Answer

(A) We know that by the $AM-GM$ inequality,for positive real numbers $a$ and $b$,$\frac{a+b}{2} \geq \sqrt{ab}$.Applying this to $2^{\sin x}$ and $2^{\cos x}$,we get:$\frac{2^{\sin x}+2^{\cos x}}{2} \geq \sqrt{2^{\sin x} \cdot 2^{\cos x}}$$\Rightarrow 2^{\sin x}+2^{\cos x} \geq 2 \cdot 2^{\frac{\sin x+\cos x}{2}} = 2^{1+\frac{\sin x+\cos x}{2}}$.We know that the range of $\sin x+\cos x$ is $[-\sqrt{2}, \sqrt{2}]$.To minimize the expression $2^{1+\frac{\sin x+\cos x}{2}}$,we must minimize the exponent $1+\frac{\sin x+\cos x}{2}$.The minimum value of $\sin x+\cos x$ is $-\sqrt{2}$.Substituting this into the expression,the minimum value is $2^{1+\frac{-\sqrt{2}}{2}} = 2^{1-\frac{1}{\sqrt{2}}}$.

The minimum value of $2^{\sin x}+2^{\cos x}$ is

Similar Questions

The number of integral values of $k$ for which the equation $3 \sin x + 4 \cos x = k + 1$ has a solution,where $k \in R$,is:

If $A>0, B>0$ and $A+B=\frac{\pi}{3}$,then the maximum value of $\tan A \tan B$ is

If $A+B+C=270^{\circ}$,then $\cos 2A+\cos 2B+\cos 2C$ is equal to:

The expression $sin^{2n}x + cos^{2n}x$ lies between which of the following values?

The minimum value of $2 \sin^2 \theta + 8 \csc^2 \theta$ is (where $\theta \in R$):-

Vedclass Test Series

Exam Paper Generator

Online Exam Module