Question

If $a=3{\cos }^{2}A+{\sin }^{4}A$, then which of the following is/are correct?

• A. The minimum value of $a$ is $1$
• B. The minimum value of $a$ is $0$
• C. The maximum value of $a$ is $3$
• D. The maximum value of $a$ is $4$

Answer 1

Answer: (A), (C)

The correct options are
A The minimum value of $a$ is $1$
C The maximum value of $a$ is $3$

Given : $a=3{\cos }^{2}A+{\sin }^{4}A$
$\Rightarrow a={\sin }^{4}A-3{\sin }^{2}A+3\Rightarrow a={({\sin }^{2}A-\frac{3}{2})}^2+\frac{3}{4}$
We know that,
$0\le {\sin }^{2}A\le 1-\frac{3}{2}\le {\sin }^{2}A-\frac{3}{2}\le -\frac{1}{2}\frac{1}{4}\le {({\sin }^{2}A-\frac{3}{2})}^2\le \frac{9}{4}1\le {({\sin }^{2}A-\frac{3}{2})}^2+\frac{3}{4}\le 3$