Question

Given $f\left(x\right)=$ $7\cos x+5\sqrt{2}\cos (x-\frac{\pi }{4})+12$ then range of $f\left(x\right)$ is

• A. $[-13,13]$
• B. $[-1,25]$
• C. $[0,25]$
• D. $[-25,25]$

Answer 1

The correct option is B $[-1,25]$

$f\left(x\right)=7\cos x+5\sqrt{2}\cos (\frac{\pi }{4}-x)+12$
$=7\cos x+5\sqrt{2}[\cos \frac{\pi }{4}\cdot \cos x+\sin \frac{\pi }{4}\cdot \sin x]+12$

$=7\cos x+5[\cos x+\sin x]+12$

$=12\cos x+5\sin x+12$

range of $12\cos x+5\sin x\in [-\sqrt{{12}^2+5^2},\sqrt{{12}^2+5^2}]$$\in [-13,13]$

$\therefore$ Range of $f\left(x\right)\in [-13+12,13+12]$ is $[-1,25]$