A) \[{{2.3}^{1/8}}\]
B) \[{{2.3}^{7/8}}\]
C) \[{{3.2}^{1/8}}\]
D) 6
Correct Answer: A
Solution :
| \[{{3}^{{{\sin }^{6}}x}}\operatorname{and}\,{{3}^{co{{s}^{6}}}}^{x}\].are positive numbers. |
| And A.M >GM \[\Rightarrow \]\[\frac{{{3}^{{{\sin }^{6}}x}}+{{3}^{{{\cos }^{6}}x}}}{2}\ge \sqrt{{{3}^{{{\sin }^{6}}x+{{\cos }^{6}}}}}\] ...(1) |
| \[=2\sqrt{{{3}^{1-\frac{3}{4}{{\sin }^{2}}2x}}}\] \[\Rightarrow \]\[{{3}^{{{\sin }^{6}}x}}+{{3}^{{{\cos }^{6}}x}}\ge 2.\sqrt{{{3}^{\frac{1}{4}}}}={{2.3}^{1/8}}\] |
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