Question

If $0<x<\frac{\pi }{2}$ then the minimum value of ${(\sin x+\cos x+cosec2x)}^3$ is

• A. 27
• B. 13.5
• C. 6.75
• D. none of these

Answer 1

The correct option is B 13.5

In the given range, the numbers $sinx,cosx$ and $cosec2x$ are all positive.
$\therefore AM\ge GM$
$\Rightarrow \frac{sinx+cosx+cosec2x}{3}\ge (sinx.cosx.cosec2x{)}^{\frac{1}{3}}$
$\Rightarrow sinx+cosx+cosec2x\ge 3(sinx.cosx.\frac{1}{2sinx.cosx}{)}^{\frac{1}{3}}=\frac{3}{2^{\frac{1}{3}}}$
$\Rightarrow (sinx+cosx+cosec2x{)}^3\ge (\frac{3}{2^{\frac{1}{3}}}{)}^3=\frac{27}{2}$
Hence, option B is correct