Question

Let $Q\left(x\right)=a{x}^2+bx+c$, where $a=c=3\sin t,b=2\cos t,t\in R-\left\{n\pi \right\}.$ If $P$ is a function of $t$ defined as $P\left(t\right)=\underset{0}{\overset{1}{\int }}Q\left(x\right)dx,$ then which of the following statement is/are correct ?

• A. Maximum value of $P\left(t\right)$ is $5$.
  
• B. $P^{\prime }\left(t\right)$ has infinitely many critical points.
• C. $P^{\prime \prime \prime }\left(t\right)\ge 0$ for all $t\in R-\left\{n\pi \right\}$
• D. $P^{\prime }\left(t\right)\ge -5$ for all $t\in R-\left\{n\pi \right\}$

Answer 1

Answer: (B), (D)

The correct options are
B $P^{\prime }\left(t\right)$ has infinitely many critical points.
D $P^{\prime }\left(t\right)\ge -5$ for all $t\in R-\left\{n\pi \right\}$

$Q\left(x\right)=a{x}^2+bx+c$
$P\left(t\right)=\underset{0}{\overset{1}{\int }}Q\left(x\right)dx$
$=\underset{0}{\overset{1}{\int }}(a{x}^2+bx+c)dx$
$={[a\frac{x^3}{3}+b\frac{x^2}{2}+cx]}_0^1$
$=\frac{a}{3}+\frac{b}{2}+c$
$=4\sin t+\cos t$
$\because a=c=3\sin t,b=2\cos t$

$P\left(t\right)=4\sin t+\cos t$
$\Rightarrow P=\sqrt{17}\sin (t+\varphi )$, where $\sin \varphi =\frac{1}{\sqrt{17}}$
$-\sqrt{17}\le P\le \sqrt{17}$