Question

The difference between the greatest and the least possible value 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

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 $=\frac{17}{4}-2=\frac{9}{4}$