Question

The absolute difference between the greatest and the least possible values of the expression $3-\cos x+{\sin }^{2}x$ is

• A. $\frac{13}{4}$
• B. $\frac{17}{4}$
• C. $\frac{9}{4}$
• D. $\frac{1}{4}$

Answer 1

The correct option is C $\frac{9}{4}$

$3-\cos x+{\sin }^{2}x$
$=-({\cos }^{2}x+\cos x)+4$
$=-{(\cos x+\frac{1}{2})}^2+\frac{17}{4}$
$\therefore$ Maximum value $=\frac{17}{4}$ at $\cos x=-\frac{1}{2}$
and Minimum value $=\frac{17}{4}-\frac{9}{4}=2$ at $\cos x=1$