QJEEM20Ma3cee: JEE Main 2020 Mathematics - Class 11 JEEM20Ma3cee Solution: The minimum value of 2 sinx + 2 cosx is :

Question

Verified Solution • 2025–26 Session moderate

The minimum value of 2^sinx + 2^cosx is :

Accepted Answer

Using AM

\geq

GM

\Rightarrow \frac{2^{\sin x} + 2^{\cos x}}{2} \geq \sqrt{2^{\sin x} {.2}^{\cos x}}

\Rightarrow 2^{\sin x} + 2^{\cos x} \geq 2^{1 + (\frac{\sin x + \cos x}{2})}

\Rightarrow min (2^{\sin x} + 2^{\cos x}) = 2^{1 - \frac{1}{\sqrt{2}}}

As we know range of sin x + cos x is :

- \sqrt{2}

\leq

sin x + cos x

\leq

\sqrt{2}

.

So Minimum value of sin x + cos x =

- \sqrt{2}

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