The point on the curve $y^2 = 16x$ for which the y-coordinate is changing 2 times as fast as the x-coordinate is :

If $y = x^3 \log x$, then $\frac{d^2 y}{dx^2}$ is equal to :

If $y = \log_e \left(\frac{x^3}{e^3}\right)$, then $\frac{d^2 y}{d x^2}$ is equal to

If $x = 2at, y = at^2$, where 'a' is a constant, then $\frac{d^2 y}{dx^2}$ at $x = 2$ is :

If $t = e^{2x}$ and $\log_e t^2$, then $\frac{d^2y}{dx^2}$ is :

For the function $f(x)=2x^3-21x^2+36x-5$, which of the following statements are correct ?

A. $x=2$ is a point of local maxima.

B. $x=6$ is a point of local minima.

C. The local maximum value is 12.

D. The local maximum value is –94.

Choose the correct answer from the options given below:

The second order derivative of which of the following functions is $5^x$ ?