find minimum value of $2^{\sin^2(\theta)}+2^{\cos^2(\theta)}$
I have found the minimum value using derivative method :
Let $f(\theta)=2^{\sin^2(\theta)}+2^{\cos^2(\theta)}$.
Then calculate $f'(\theta)$ and $f''(\theta)$.
Is it possible to find minimum value by alternative process without using the concept of derivative?
Best Answer
HINT:
For real $a>0,$
$$(a^2)^{\sin^2\theta}+(a^2)^{\cos^2\theta}=a\left(a^{-\cos2\theta}+a^{\cos2\theta}\right)$$
Now $\dfrac{a^{-\cos2\theta}+a^{\cos2\theta}}2\ge\sqrt{a^{-\cos2\theta}\cdot a^{\cos2\theta}}=1$
Can you identify $a$ here?