Find all relative extrema of the function. Use a graphing utility to verify your result. (If an answer does not exist, enter DNE.)

Question

Find all relative extrema of the function. Use a graphing utility to verify your result. (If an answer does not exist, enter DNE.)

y = ln(3 + 9x^2 − x^3)

Answer 1

Differentiate y = ln(3 + 9x² - x³) using the chain rule:

Let u = 3 + 9x² - x^{3 du}/_dx = 18x - 3x²

Then y = ln(u) ^dy/_du = ¹/_u

^dy/_dx = ^dy/_du x ^du/_dx = ¹/_u x (18x - 3x²) = 1/(3 + 9x² - x³)(18x - 3x²)

For relative extrema, ^dy/_dx = 0

1/(3 + 9x² - x³)(18x - 3x²) = 0

1/(3 + 9x² - x³) ≠ 0

so solve 18x - 3x² = 0

6x - x² = 0

x(6 - x) = 0 Therefore relative extrema are at x = 0 and x = 6

Plug these values into the original function:

At x = 0, y = ln(3) = 1.099

At x = 6, y = ln(111) = 4.710