Let `y=3x.(x+a)/(x+b)+5` [a and b are positive constants and `a gt b`] be the total cost for x unit of output of a commodity. Show that the marginal cost falls continuosly as the output increases.

Let y=3x. (x+7)/(x+5)+4 be the total cost for x units of output of a product. Show that the marginal cost falls continuously as the output increases. [Note that the marginal cost of a product is the rate of change in cost for unit change in the output.]

If y is the total cost of x units of output and it is given that marginal cost equals average cost, show that the average cost function is constant.

If a and b are constants, then show that var (ax+b)=a^2 var (x) .

If a, b, c and x, y, z are positive, then show that (a/x + b/y + c/z) (x/a + y/b + z/c) ge 9

If the marginal function for x units of output is given by (6)/((x+2)^(2))+5 , find the total revenue function and the demand law.

If x gt 1/2 , show that the function f(x) =x(4x^(2)-3) is steadily increasing.

A firm produces x units of output per week at a total cost of Rs. ((1)/(3)x^(3)-x^(2)+5x+3) .Find the output levels at which the marginal cost, the average variable cost attain their respective minima.

One kg of food X contains 6 units of Vitamin A and 7 units of Vitamin B. One kg of food Y contains 8 units of Vitamin A and 12 units of Vitamin B. Cost of each kg of food X and food Y are Rs. 12 and Rs. 20 respectively. The daily minimum requirement of Vitamin A and Vitamin B are 100 units and 120 units respectively. How much food X and food Y are to be mixed so that the cost will be minimum? Formulate the problem as a linear programming problem.

Let f(x) = a+ b|x| + c|x|^4 where a, b and continuous are real constants . Then, f(x) is differentiable at x = 0, if

Let f(x)= { x e^(a x),x lt=0;x+a x^2-x^3,x >0 where a is a positive constant. Find the interval in which f^(prime)(x) is increasing.