If $f(x)=\int_0^x e^{t^2}(t-2)(t-3) dt$ for all $x \in(0, \infty)$,then
$(A)$ $f$ has a local maximum at $x=2$
$(B)$ $f$ is decreasing on $(2,3)$
$(C)$ there exists some $c \in(0, \infty)$ such that $f^{\prime \prime}(c)=0$
$(D)$ $f$ has a local minimum at $x=3$

$f(x)=4 \log _{e}(x-1)-2 x^{2}+4 x+5, x>1$,which one of the following is $NOT$ correct?

Consider the polynomial $f(x)=1+2x+3x^2+4x^3$. Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t=|s|$.
$1.$ The real number $s$ lies in the interval
$(A)$ $\left(-\frac{1}{4}, 0\right)$ $(B)$ $\left(-1,-\frac{3}{4}\right)$
$(C)$ $\left(-\frac{3}{4},-\frac{1}{2}\right)$ $(D)$ $\left(0, \frac{1}{4}\right)$
$2.$ The area bounded by the curve $y=f(x)$ and the lines $x=0, y=0$ and $x=t$,lies in the interval
$(A)$ $\left(\frac{3}{4}, 3\right)$ $(B)$ $\left(\frac{21}{64}, \frac{11}{16}\right)$
$(C)$ $(9,10)$ $(D)$ $\left(0, \frac{21}{64}\right)$
$3.$ The function $f^{\prime}(x)$ is
$(A)$ increasing in $\left(-t,-\frac{1}{4}\right)$ and decreasing in $\left(-\frac{1}{4}, t\right)$
$(B)$ decreasing in $\left(-t,-\frac{1}{4}\right)$ and increasing in $\left(-\frac{1}{4}, t\right)$
$(C)$ increasing in $(-t, t)$ $(D)$ decreasing in $(-t, t)$
Give the answer for questions $1, 2$ and $3.$

Two differentiable functions $f(x)$ and $g(x)$ are such that $f''(x) > 0$ and $g''(x) < 0$ for all $x \in (a,b)$ and $\int_{a}^{b} f(x) dx = \int_{a}^{b} g(x) dx$. If $f(x) = g(x)$ for $x = \alpha, \beta \in (a,b)$ $(\alpha < \beta)$,then:

For the function $f(x) = x^4 (12 \ln x - 7)$,which of the following statements is true?

Let $f(x) = \sin x + (x^3 - 3x^2 + 4x - 2) \cos x$ for $x \in (0, 1)$. Consider the following statements:
$I.$ $f$ has a zero in $(0, 1)$.
$II.$ $f$ is monotone in $(0, 1)$.
Then,