Question

The minimum value of $27\sec x+64\text{cosec}x$ for $x\in (0,\pi /2)$ is

• A. $5$
• B. $25$
• C. $125$
• D. $625$

Answer 1

The correct option is C $125$

Let $f\left(x\right)=27\sec x+64\text{cosec}x$
$\Rightarrow f^{\prime }\left(x\right)=27\sec x\tan x-64\text{cosec}x\cot x$
for minimum or maximum value
$f^{\prime }\left(x\right)=0$
$\Rightarrow \frac{3^3}{4^3}={\cot }^{3}x\Rightarrow \cot x=\frac{3}{4}\Rightarrow \sec x=\frac{5}{3},\text{cosec}x=\frac{5}{4}$
$\therefore f^{\prime }(x{)}_{\text{mini}}=45+80=125$