For all $a, b \in R$,the function $f(x) = 3x^4 - 4x^3 + 6x^2 + ax + b$ has:

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 $X_f = 1$

$A$ wire of length $2$ units is cut into two parts,which are bent respectively to form a square of side $x$ units and a circle of radius $r$ units. If the sum of the areas of the square and the circle so formed is minimum,then:

The number of points where the curve $y=x^5-20x^3+50x+2$ crosses the $x$-axis is $............$.

Find two numbers whose sum is $24$ and whose product is as large as possible.

The maximum value of $f(x) = x^2 \log x$ on the interval $[1, e]$ is: