Trigonometric Equations Question 84
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
Show Answer
Answer:
Correct Answer: A
Solution:
- Since A.M. $ \ge $ G.M. $ \frac{1}{2}({2^{\sin x}}+{2^{\cos x}})\ge \sqrt{{2^{\sin x}}{{.2}^{\cos x}}} $
$ \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
