Need some quick help on a homework problem

Question

Hey,

Need some quick help with a homework problem. I found the derivative for part a but I'm not sure how to do part b.

The cost for a company to produce x pairs of jeans is given by 𝐶(𝑥)=2000+3𝑥+0.01x^2+0.0002x^3.On any given day the company produces [100,400]pairs of jeans.

(a) Find the derivative of 𝐶(𝑥). This is called the marginal cost.

(b) Use MVT to find when the marginal cost equals the average cost.

Answer 1

One way of calculating the average cost is [C(400) - C(100)]/(400 - 100) = 50

The mean value theorem (MVT) says that as long as the cost function is continuous and differentiable on the interval, that for at least on place on the interval, the slope at that point will equal the average on the interval.

The derivative is C'(x) = 3 + 0,02x + 0.0006x²

So, per the MVT, 3 + 0,02x + 0.0006x² = 50

0.0006x² + 0,02x - 47 = 0

Using the quadratic formula, we can solve for x.

x = 263.71 and x = -297.04

We can throw out x = -297.04 and round to the nearest whole number to get x = 264

So, when producing 264 pairs of jeans, the cost per pair of jeans equals the average cost per pair of jeans between 100 and 400 pairs.

Answer 2