Question

For x, t $ϵ$ R let
$p_1\left(x\right)=\left(\sin t\right)x^2-\left(2\cos t\right)x+\sin t$
be a family of quadratic polynomials in x with variable co efficients. Let A (t) = ${\int }_0^1{p}_t\left(x\right)dx$ . Which of the following statements are true?
(I) A (t) < 0 for all t.
(II) A (t) has infinitely many critical points.
(III) A (t) = 0 for infinitely many t.
(IV) A'(t) < 0 for all t.

• A. (I) and (II) only
• B. (II) and (III) only
• C. (III) and (IV) only
• D. (IV) and (I) only

Answer 1

Answer: (B)

(II) and (III) only

Given $p_t\left(x\right)=\left(\sin t\right)x^2-\left(2\cos t\right)x+\sin t$

$A\left(t\right)={\int }_0^1{p}_t\left(x\right)dx=\sin t\left({\int }_0^1{x}^{2}dx\right)-2\cos t\left({\int }_0^{1}xdx\right)+\sin t=\frac{4}{3}\sin t-\cos t$

let $\cos \varphi =\frac{4}{5}$ and $\sin \varphi =\frac{3}{5}$

$\Rightarrow A\left(t\right)=\frac{5}{3}(\sin t\cos \varphi -\cos t\sin \varphi )=\frac{5}{3}\sin (t-\varphi )$

Clearly, $-\frac{5}{3}\le A\left(t\right)\le \frac{5}{3}$

and $A\left(t\right)$ has infinitely many zeros and critical points.

Also, $A\left(t\right)=0⟹\sin (t-\varphi )=0$

So, $A\left(t\right)=0$ for infinitely many $t$.