`f(x) = x^3 + 6x^2 - 15x` Find the critical numbers of the function

Textbook Question

Chapter 4, 4.1 - Problem 30 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the critical numbers of the function, hence, you need to evaluate the soutions to the first derivative, such that:

`f'(x) = 0`

`f'(x) = (x^3 + 6x^2 - 15x)`

`f'(x) = 3x^2 + 12x - 15`

You need to solve for x the equation f'(x) = 0:

`3x^2 + 12x - 15 = 0`

You need to divide by 3:

`x^2 + 4x - 5 = 0`

Using quadratic formula, yields:

`x_(1,2) = (-4+-sqrt(16 + 20))/2 => x_(1,2) = (-4+-sqrt36)/2`

`x_(1,2) = (-4+-6)/2 => x_1 = (6-4)/2 => x_1 = 1`

`x_2 = (-4-6)/2 => x_2 = -5`

Hence, evaluating the critical values of the function yields `x = 1, x = -5.`

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