If $x = 2t^2 + 3, y = 3t^2 + 6t + 5$, then the value of $\frac{d^2y}{dx^2}$ is :

Let $f : R \to R$ be defined such that $f(x) = 16x^2 - 16x + 12$

(A) Maximum value of $f(x)$ is 8

(B) Minimum value of $f(x)$ is 8

(C) Minimum value of $f(x)$ is 16

(D) No maximum value of $f(x)$

Choose the correct answer from the options given below :

The order and degree Of the differential equation $\frac{d^2 y}{dx^2} + 2 e^{-x} = 0$, respectively are

The value of the integral $\int e^x \left(\frac{1}{x} - \frac{1}{x^2}\right) dx$ is :

If $x^{2/3} + y^{2/3} = a^{2/3}$, then $\frac{dy}{dx}$ is equal to :

If $R(x)$ is the total revenue received and $C(x)$ is the total cost incurred in the production of x units of a commodity, then the profit function $P(x)$

If $v = \frac{4}{3} \pi r^3$, at what rate is cubic / unit sec is increasing when $r = 10$, and $\frac{dr}{dt} = 0.01$ ?