Maxima and Minima

Find the absolute maximum value and the absolute minimum value of the following function in the given interval:

f (x) = sin x + cos x , x ∈ [0, π]

A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.

A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum

Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.

Prove that the following function do not have maxima or minima:

f(x) = e^x

Find the maximum and minimum of the following functions : f(x) = `logx/x`

Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.

Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:

f(x) `= x sqrt(1 - x), 0 < x < 1`