Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be:

f(x) = sin x – cos x, 0 < x < 2π

Show that semi-vertical angle of right circular cone of given surface area and maximum volume is .

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is of the volume of the sphere.

Find two positive numbers x and y such that their sum is 35 and the product x² y⁵ is a maximum.

Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is .

Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?

Find the maximum and minimum values of x + sin 2x on [0, 2π].

It is given that at x = 1, the function x⁴ – 62x² + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.

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

At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?