Question

The minimum value of $f\left(x\right)=4{\sec }^{2}x+9{\text{cosec}}^{2}x$ is

• A. $5$
• B. $25$
• C. $13$
• D. ${13}^2$

Answer 1

Answer: (B)

$25$

$f\left(x\right)=13+4{\tan }^{2}x+9{\cot }^{2}x$
Use AM & G.M
$\frac{4{\tan }^{2}x+9{\cot }^{2}x}{2}\ge (4{\tan }^{2}x\times 9{\cot }^{2}x{)}^{\frac{1}{2}}$
$4{\tan }^{2}x+9{\cot }^{2}x\ge 12$
$13+4{\tan }^{2}x+9{\cot }^{2}x\ge 25$
$f\left(x\right)\ge 25$