Question

The range of the function $f\left(x\right)=5\cos x+3\cos (x+\pi /3)+4$ is :

• A. $[-3,11]$
• B. $[-18,10]$
• C. $[-10,18]$
• D. $Noneofthese$

Answer 1

The correct option is A $[-3,11]$

$A]5\cos x+3\cos (x+\frac{\pi }{3})+4$

$=5\cos x+3(\cos x\cos \frac{\pi }{3}-\sin x\sin \frac{\pi }{3})+4$

$-5\cos x+3(\frac{\cos x}{2}-\sin x\frac{\sqrt{3}}{2})+4$

$=\frac{13}{2}\cos x-\frac{3\sqrt{3}}{2}\sin x+4$

Let $\frac{13}{2}=a\frac{3\sqrt{3}}{2}=b$

$=a\cos x-b\sin x+4$

max value $=\sqrt{a^2+b^2}+4$

$=\sqrt{{\left(\frac{13}{2}\right)}^2+{\left(\frac{3\sqrt{3}}{2}\right)}^2}+4$

$=\sqrt{\frac{169+27}{4}}+4=\sqrt{\frac{196}{4}}+4=\frac{14}{2}+4$

$=\frac{22}{2}=11$

min value $-(\frac{14}{2}-4)=-3$