Show that the function f(x) = 4x³ – 18x² + 27x – 7 is always increasing on R.

Differentiate the function (sin x)^x + sin^–1 √xwith respect to x.

If x^m yⁿ = (x + y)^{m + n}, prove that 

If the sum of lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum, when the angle between them is π/3.

Solve the differential equation 

The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x³ - 0.02 x² + 30 x +5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of total cost at any level of output.

417.

Differentiate ${\tan }^{-1} \left(\frac{1 + \cos \mathrm{x}}{\sin \mathrm{x}}\right)$ with respect x.

Find the equations of the tangent and the normal, to curve 16x² + 9y² = 145 at the point (x₁,y₁) where x₁ = 2 and y₁ > 0

Find the intervals in which the function f(x)  =  is

a) Strictly increasing

b) Strictly decreasing

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question?