Suppose that the cost (in dollars) for a company to produce x pairs of a new line of jeans is given byC(x)=2000+3x+0.01x^2 +0.0002x^3(a) find the marginal cost funtion(b) findC'(100) and explain its meaning. What does it predict?

Answer:The marginal cost function is [tex]C'(x)=0.0006x^2+0.02x+3[/tex]The marginal cost at the production level of 100 pairs is[tex]C'(100)=\$11/pair[/tex]. This gives the rate at which costs are increasing with respect to the production level when x = 100 and predicts the cost of the 101st pair of jeans.Step-by-step explanation:(a) The marginal cost function is the derivative of the cost function. [tex]\:{Marginal \:Cost} = \frac{d}{dx}(C(x))=C'(x)\\\\C'(x)=\frac{d}{dx}(2000+3x+0.01x^2 +0.0002x^3)\\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\C'(x)=\frac{d}{dx}\left(2000\right)+\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(0.01x^2\right)+\frac{d}{dx}\left(0.0002x^3\right)\\\\C'(x)=0.0006x^2+0.02x+3[/tex](b) The marginal cost at the production level of 100 pairs is[tex]C'(x)=0.0006x^2+0.02x+3\\C'(100)= 0.0006(100)^2+0.02(100)+3\\C'(100)=\$11/pair[/tex]This gives the rate at which costs are increasing with respect to the production level when x = 100 and predicts the cost of the 101st pair of jeans.