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Relative extrema, production level

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See the attached file for the problem.

1. Find all relative extrema. Use the Second Derivative Test if applicable.
g(x) = x^2(6-x)^3

2. A manufacturer has determined that the total cost C of operating a factory is C = 0.5x^2 + 15x + 5000, where x is the number of units produced. At what level of production will the average cost be minimized? (The average cost per unit is C/x.)

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Solution Summary

The solution shows how to find all relative extrema by the Second Derivative Test. Moreover, it provides an real world application where the average cost is minimized. Step by step calculations are given, along with explanations for each step.

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Please help on these two problems.

9. Find all relative extrema. Use the Second Derivative Test if applicable.

g(x) = x^2(6-x)^3

First find the derivatives:
Applying the chain rule:
g'(x) = 2x(6-x)^3 + 3x^2(6-x)^2(0-1)
g'(x) = 2x(6-x)^3 - 3x^2(6-x)^2

The second derivative is
g"(x) = 2(6-x)^3 + 2x*3(6-x)^2(0-1) - 3*2x(6-x)^2 - 3x^2*(6-x)(0-1)
g"(x) = 2(6-x)^3 - ...

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