Question

The difference between the greatest and the least possible value of the expression $3-cosx+si{n}^{2}x$ is

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

Answer 1

Answer: (C)

$\begin{array}{l}3-\cos x+1-{\cos }^{2}x\\ =-\left({\cos }^{2}x+\cos x\right)+4\\ =-\left({\cos }^{2}x+\cos x+\frac{1}{4}\right)+4+\frac{1}{4}\\ =-{\left(\cos x+\frac{1}{2}\right)}^2+\frac{17}{4}\\ =\frac{17}{4}-{\left(\cos x+\frac{1}{2}\right)}^2\end{array}$
Maximum value (at $\cos x=-\frac{1}{2}$ ) $=\frac{17}{4}$
Minimum value (at $\cos x=1$ ) $=\frac{17}{4}-\frac{9}{4}=2$
Difference $=\frac{17}{4}-2=\frac{9}{4}$