KVPY 2013 - 10

For $x,t\in\mathbb{R}$ , let $p_t(x)=(sint)x^2-(2cost)x+sint$

be a family of quadratic polynomials in $x$ with variable coefficients.

Let $A(t)=\int_{0}^{1}p_t(x)dx$

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 $t$

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Try the whole set KVPY 2013 SB/SX here.