The abscissae of the points of the curve $y=x^3$ are in the interval $[-2,2]$, where the slope of the tangents can be obtained by mean value theorem for the interval $[-2,2]$ are

Let $x$ be the length of each of the equal sides of an isosceles triangle and $\theta$ be the angle between these sides. If $x$ is increasing at the rate $\frac{1}{12} \mathrm{~m} /$ hour and $\theta$ is increasing at the rate $\frac{\pi}{180} \mathrm{rad} /$ hour, then the rate at which area of the triangle is increasing when $x=12 \mathrm{~m}$ and $\theta=\frac{\pi}{4}$ is

A wire of length 8 units is cut into two parts which are bent respectively in the form of a square and a circle. The least value of the sum of the areas so formed is

The radius of the base of a cone is increasing at the rate $3 \mathrm{~cm} /$ minute and the altitude is decreasing at the rate $4 \mathrm{~cm} /$ minute . The rate at which the lateral surface area is changing, when the radius is 7 cm and altitude is 24 cm is