Suppose $x_1$ and $x_2$ are the point of maximum and the point of minimum respectively of the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$. If the equality $x_1^2 = x_2$ holds true,then the value of $a$ must be:

The absolute maximum value of the function $f(x) = 2x^3 - 3x^2 - 36x + 9$ defined on $[-3, 3]$ is

The minimum value of the function $f(x) = \frac{40}{3x^4 + 8x^3 - 18x^2 + 60}$ is:

For the function $f(x) = x \cos \frac{1}{x}, \quad x \geq 1$,consider the following statements:
$(A)$ For at least one $x$ in the interval $[1, \infty), f(x+2)-f(x) < 2$
$(B)$ $\lim _{x \rightarrow \infty} f^{\prime}(x) = 1$
$(C)$ For all $x$ in the interval $[1, \infty), f(x+2)-f(x) > 2$
$(D)$ $f^{\prime}(x)$ is strictly decreasing in the interval $[1, \infty)$
Which of the following combinations of statements is correct?

The displacement $x$ of a particle at time $t$ is given by $x = t^4 - kt^3$. If the velocity of the particle is maximum at $t = 2$,then $k = $ ..........

If $f(x) = ax + \frac{b}{x}$ where $a, b, x > 0$,then the value of $x$ for which $f(x)$ is minimum is: