The function $f(x)=\frac{x}{2}+\frac{2}{x}$ has a local minima at $x$ equal to

Given a curve $y=7 x-x^3$ and $x$ increases at the rate of 2 units per second. The rate at which the slope of the curve is changing, when $x=5$ is

The derivative of $$x^{2 x}$$ w.r.t. $$x$$ is:

The interval in which the function $$f(x)=2 x^3+9 x^2 +12 x-1$$ is decreasing is :

The area of the circle is increasing at a uniform rate of $2 \mathrm{~cm}^2 / \mathrm{s}$. How fast is the circumference of the circle increasing when the radius $r=5 \mathrm{~cm}$ ?

If $x=a \sin ^3 \theta, y=b \cos ^3 \theta$,then find $\frac{d^2 y}{d x^2}$ at $\theta=\frac{\pi}{4}$.

37. A rectangular visiting card is to contain $24 \mathrm{sq} . \mathrm{cm}$. of printed matter. The margins at the top and bottom of the card are to be 1 cm and the margins on the left and right are to be $1 \frac{1}{2} \mathrm{~cm}$ as shown below:

On the basis of the above information, answer the following questions:

(i) Write the expression for the area of the visiting card in terms of $x$.

(ii) Obtain the dimensions of the card of minimum area.

(a) The median of an equilateral triangle is increasing at the rate of $$2 \sqrt{3} \mathrm{~cm} / \mathrm{s}$$. Find the rate at which its side is increasing.

OR

(b) Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.