A wire of length $28$ m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
(iii) $h(x) = \sin x + \cos x, 0 < x < \frac{\pi}{2}$

Show that the function given by $f(x) = 3x + 17$ is increasing on R.

Prove that the function $f$ given by $f(x) = \log |\cos x|$ is decreasing on $(0, \frac{\pi}{2})$ and increasing on $(\frac{3\pi}{2}, 2\pi)$.

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
(iii) $f(x) = 4x - \frac{1}{2}x^2, x \in [-2, \frac{9}{2}]$

A balloon, which always remains spherical on inflation, is being inflated by pumping in $900$ cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is $15$ cm.

The length $x$ of a rectangle is decreasing at the rate of $5$ cm/minute and the width $y$ is increasing at the rate of $4$ cm/minute. When $x = 8$cm and $y = 6$cm, find the rates of change of (b) the area of the rectangle.

The maximum value of $[x(x-1)+1]^{\frac{1}{3}}$, $0 \le x \le 1$ is

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
(iv) $f(x) = \sin x – \cos x, 0 < x < 2\pi$

A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is $10$ cm.

Find the maximum and minimum values, if any, of the following functions given by
(iv) $g(x) = x^3 + 1$

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
(ii) $f(x) = \sin x + \cos x, x \in [0, \pi]$

Find two positive numbers $x$ and $y$ such that their sum is $35$ and the product $x^2 y^5$ is a maximum.

Sand is pouring from a pipe at the rate of $12$ cm$^3$/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is $4$ cm?

Prove that the volume of the largest cone that can be inscribed in a sphere of radius $R$ is $\frac{8}{27}$ of the volume of the sphere.

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
(iv) $f(x) = (x-1)^2 + 3, x \in [-3, 1]$

Find the intervals in which the following functions are strictly increasing or decreasing:
(d) $6 – 9x – x^2$

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
(ii) $g(x) = x^3 – 3x$

Find two positive numbers whose sum is $16$ and the sum of whose cubes is minimum.

A ladder $5$ m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of $2$cm/s. How fast is its height on the wall decreasing when the foot of the ladder is $4$ m away from the wall ?