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प्रश्न
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
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उत्तर
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")`
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