If x is a real number,what is the minimum val | vedclass

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

Question

(A) Given the function $f(x) = 3^{x+1} + 3^{-(x+1)}$.We can rewrite this as $f(x) = 3^{x+1} + \frac{1}{3^{x+1}}$.Since $3^{x+1} > 0$ for all real $x$,

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(A) Given the function $f(x) = 3^{x+1} + 3^{-(x+1)}$.We can rewrite this as $f(x) = 3^{x+1} + \frac{1}{3^{x+1}}$.Since $3^{x+1} > 0$ for all real $x$,we can apply the Arithmetic Mean-Geometric Mean $(AM \geq GM)$ inequality.For any two positive numbers $a$ and $b$,$\frac{a+b}{2} \geq \sqrt{ab}$,which implies $a+b \geq 2\sqrt{ab}$.Let $a = 3^{x+1}$ and $b = \frac{1}{3^{x+1}}$.Then $f(x) = a + b \geq 2\sqrt{a \cdot b}$.$f(x) \geq 2\sqrt{3^{x+1} \cdot \frac{1}{3^{x+1}}}$.$f(x) \geq 2\sqrt{1}$.$f(x) \geq 2$.Thus,the minimum value of the function is $2$.

If $x$ is a real number,what is the minimum value of $f(x) = 3^{x+1} + 3^{-(x+1)}$?

Similar Questions

The value of $k$,for which $(\cos x + \sin x)^2 + k \sin x \cos x - 1 = 0$ is an identity,is

The smallest value of $5 \cos \theta + 12$ is

The maximum value of $\cos^2 \left( \frac{\pi}{3} - x \right) - \cos^2 \left( \frac{\pi}{3} + x \right)$ is

For $A, B$ and $C$,if $A+B+C=0$,then $\sin(2A) + \sin(2B) + \sin(2C)$ is equal to

If $u = \sqrt{a^2 \cos^2 \theta + b^2 \sin^2 \theta} + \sqrt{a^2 \sin^2 \theta + b^2 \cos^2 \theta}$,then what is the difference between the maximum and minimum values of $u^2$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module