Number of solutions of the equation $3 \tan x + x^3 = 2$ in the interval $(0, \frac{\pi}{4})$ is:

If the sum of two numbers is $3$,then the maximum value of the product of the first number and the square of the second number is:

The function $S(x) = \int\limits_0^x {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)\,dt} $ has two critical points in the interval $[1, 2.4]$. One of the critical points is a local minimum and the other is a local maximum. The local minimum occurs at $x =$

If the function $f(x)=2x^{3}-9ax^{2}+12a^{2}x+1$ where $a>0$ attains its maximum and minimum at $p$ and $q$ respectively such that $p^{2}=q$,then $a$ is equal to

The maximum value of $f(x) = (7-x)^4 (2+x)^5$ is

Let $S$ be the set of all twice differentiable functions $f$ from $R$ to $R$ such that $\frac{d^2 f}{d x^2}(x) > 0$ for all $x \in (-1, 1)$. For $f \in S$,let $X_f$ be the number of points $x \in (-1, 1)$ for which $f(x) = x$. Then which of the following statements is(are) true?
$(A)$ There exists a function $f \in S$ such that $X_f = 0$
$(B)$ For every function $f \in S$,we have $X_f \leq 2$
$(C)$ There exists a function $f \in S$ such that $X_f = 2$
$(D)$ There does $NOT$ exist any function $f$ in $S$ such that $X_f = 1$