Question

The range of the function $f\left(x\right)=({\cos }^{2}x+4{\sec }^{2}x)$ is

• A. $[4,\infty )$
• B. $[0,\infty )$
• C. $[5,\infty )$
• D. $(0,\infty )$

Answer 1

The correct option is A $[4,\infty )$

As we know that

$AM\ge GM$

$⟹\frac{{\cos }^{2}x+{\text{sec}}^{2}x}{2}\ge \sqrt{\left({\cos }^{2}x\right)\left(4{\text{sec}}^{2}x\right)}$

$⟹\frac{f\left(x\right)}{2}\ge \sqrt{4}$

$⟹f\left(x\right)\ge 4$

So the range of $f\left(x\right)$ is $[4,\infty )$