Question

If the marginal cost function of x units of output is `"a"/sqrt("a"x + "b")` and if the cost of output is zero. Find the total cost as a function of x

Answer 1

M.C = `"a"/sqrt("a"x + "b")`

Total cost function

C = `int ("M.C")  "d"x`

C = `int "a"("a"x + "b")^(1/2)  "d"x`

= `int "a"("a"x + "b")^((-1)/2)  "d"x + "k"`

= `"a"[(("a"x + "b")^((-1)/2) + 1)/(((-1)/2 + 1) xx ("a"))] + "k"`

C = `[("a"x + "b")^(1/2)/((1/2))] + "k"`

∴ C(x) = `2("a"x + "b")^(1/2)`  ........(1)

When x = 0

Equation (1)

⇒ 0 = `2["a"(0) + "b"]^(1/2) + "k"`

k = `-2("b")^(1/2)`

⇒ k = `-2sqrt("b")`

Required cost function

C(x) = `2("a"x + "b")^(1/2) - 2sqrt("b")`

∴ C = `2sqrt("a"x + "b") - 2sqrt("b")`