Question

If $f\left(x\right)={\cos }^{2}x+se{c}^{2}x$, then

• A. $f\left(x\right)<1$
• B. $1<f\left(x\right)<2$
• C. $f\left(x\right)$greater than or equal to $2$
• D. $f\left(x\right)=1$

Answer 1

The correct option is C

$f\left(x\right)$greater than or equal to $2$

Explanation for the correct option:

Use the concept of $AM-GM$

The arithmetic means (AM) is greater than equal to the geometric mean (GM).

$\therefore AM\ge GM$

$\begin{array}{l}\Rightarrow \frac{{\cos }^{2}x+{\sec }^{2}x}{2}\ge \sqrt{{\cos }^{2}x{\sec }^{2}x}\\ \Rightarrow \frac{{\cos }^{2}x+{\sec }^{2}x}{2}\ge \sqrt{{\cos }^{2}x\frac{1}{{\cos }^{2}x}}\left[\because \sec \theta =\frac{1}{\cos \theta }\right]\\ \Rightarrow \frac{{\cos }^{2}x+{\sec }^{2}x}{2}\ge \sqrt{1}\end{array}$

$\begin{array}{l}\Rightarrow \frac{{\cos }^{2}x+{\sec }^{2}x}{2}\times 2\ge 1\times 2\\ \Rightarrow {\cos }^{2}x+{\sec }^{2}x\ge 2\\ \Rightarrow f\left(x\right)\ge 2\end{array}$

Hence, the correct option is (C).