Find minimum value of 2^sin x + 2^cos x?

How to do this? Any particular method?

I tried finding dy/dx and setting it to zero. That gave me a relation (2^sin x)/(2^cos x) = tan x(I couldnt find out x from this). Please help?

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- Aug 28th 2008, 06:34 AMfardeen_genFind minimum value of...?
Find minimum value of 2^sin x + 2^cos x?

How to do this? Any particular method?

I tried finding dy/dx and setting it to zero. That gave me a relation (2^sin x)/(2^cos x) = tan x(I couldnt find out x from this). Please help? - Aug 28th 2008, 03:59 PMmr fantastic
- Aug 28th 2008, 07:26 PMfardeen_gen
I don't.

- Aug 28th 2008, 11:37 PMwingless
Do you see that this function is periodic?

$\displaystyle f(x) = 2^{\sin x} + 2^{\cos x}$

$\displaystyle f(x) = f(x+2\pi) ~\forall x$ - Aug 29th 2008, 12:02 AMwingless
$\displaystyle f(x) = 2^{\sin x} + 2^{\cos x}$

$\displaystyle f'(x) = 0 ~\rightarrow~ 2^{\sin x}\cos x = 2^{\cos x}\sin x$

One obvious attempt to solve this equation is $\displaystyle \sin x = \cos x$. The solution set to this equation is $\displaystyle x = \left \{ \frac{\pi}{4}, \frac{\pi}{4}+\pi\right \}$. One of these is maximum and the other is minimum. Note that there are infinitely many maximums and minimums, as the function has a period of $\displaystyle 2\pi$. - Aug 29th 2008, 01:33 AMMoo