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

The total cost function for $x$ units of a commodity is given by $C(x) = \frac{25x^3}{3} - 75x^2 + 48x + 34$. The output $x$ at which the marginal cost is minimum is :

Two positive numbers $x$ and $y$ such that $x + y = 60$ and $xy^3$ is maximum are :

If $f(x) = a\log|x| + \dfrac{b}{x} + x$ has its extreme values at $x = -1$ and $x = 3$, then $(a, b)$ is equal to:

The demand function P for maximising a profit monopolist is given by P=274-x², while the marginal cost is 4+3x for x units of commodity. The consumer surplus is

The degree of the differential equation $\left(1 - \left(\frac{dy}{dx}\right)^2\right)^{3/2} = k \frac{d^2 y}{dx^2}$ is :

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