Trigonometric-Equations Question 395
Question: The only value of x for which $ {2^{\sin x}}+{2^{\cos x}}>{2^{1-(1/\sqrt{2})}} $ holds, is
Options:
A) $ \frac{5\pi }{4} $
B) $ \frac{3\pi }{4} $
C) $ \frac{\pi }{2} $
D) All values of x
Correct Answer: A
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Answer:
Solution:
$ \Rightarrow $ $ {2^{\sin x}}+{2^{\cos x}}\ge {{2.2}^{\frac{\sin x+\cos x}{2}}} $
$ \Rightarrow $ $ {2^{\sin x}}+{2^{\cos x}}\ge {2^{1+\frac{\sin x+\cos x}{2}}} $ and we know that $ \sin x+\cos x\ge -\sqrt{2} $
$ \therefore $ $ {2^{\sin x}}+{2^{\cos x}}>{2^{1-(1/\sqrt{2})}} $ , for $ x=\frac{5\pi }{4} $
BETA
