The curve $y = 2x^3 + ax^2 + bx + c$ passes through the origin,and the tangents at $x = -1$ and $x = 2$ are parallel to the $X$-axis. Then the values of $a, b,$ and $c$ are respectively:

If $y = a \ln x + bx^2 + x$ has its extremum value at $x = 1$ and $x = 2,$ then $(a, b) =$

Let a function $f(x)$ be defined as $f(x) = \begin{cases} \cos^{-1}(\mu) + x^2, & 0 < x < 1 \\ 4x, & x \geqslant 1 \end{cases}$. The function $f(x)$ can have a local minimum at $x = 1$ if the value of $\mu$ lies in the interval:

The area (in sq. units) of the largest rectangle $ABCD$ whose vertices $A$ and $B$ lie on the $x$-axis and vertices $C$ and $D$ lie on the parabola $y = x^{2}-1$ below the $x$-axis,is

Two positive numbers $x$ and $y$ are such that $(x+y)=60$ and $x y^3$ is maximum. Then the numbers $x$ and $y$ are respectively:

For every twice differentiable function $f : R \rightarrow [-2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$,which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ There exist $r, s \in R$,where $r < s$,such that $f$ is one-one on the open interval $(r, s)$.
$(B)$ There exists $x_0 \in (-4, 0)$ such that $|f'(x_0)| \leq 1$.
$(C)$ $\lim_{x \rightarrow \infty} f(x) = 1$.
$(D)$ There exists $a \in (-4, 4)$ such that $f(a) + f''(a) = 0$ and $f'(a) \neq 0$.