Question

The maximum value of $F\left(x\right)={\int }_{7\pi /6}^x(a\sin t+b\cos t)dt,$$a,b\in R^{+},x\in [\frac{7\pi }{6},\sqrt{21}]$ is

• A. $a+b$
• B. $a\sqrt{21}-b\frac{7\pi }{6}$
• C. $0$
• D. None of these

Answer 1

The correct option is C $0$

$F\left(x\right)={\int }_{7\pi /6}^x(a\sin t+b\cos t)dt$

$\therefore F^{\prime }\left(x\right)=a\sin x+b\cos x<0\forall x\in [\pi ,\frac{3\pi }{2}]$

$\Rightarrow F\left(x\right)$ is decreasing in $[\frac{7\pi }{6},\sqrt{21}]$

So, $F\left(x\right)$ is maximum at $x=\frac{7\pi }{6}$ which is $0$