Question

Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x)=x³ +3 x² -105 x (x=___)

Answer 1

The critical numbers of the function f(x) = x³ + 3x² - 105x are -7 and 5.

Step-by-step explanation:

The critical numbers of a function are the values of x where the derivative of the function is equal to zero or does not exist. To find the critical numbers of the function f(x) = x³ + 3x² - 105x, we need to find the derivative of f(x) and solve for x when the derivative is equal to zero:

f'(x) = 3x² + 6x - 105

To find the critical numbers, we set f'(x) = 0 and solve for x:

3x² + 6x - 105 = 0

This is a quadratic equation that can be factored or solved using the quadratic formula. However, in this case, the equation can be factored as (3x + 21)(x - 5) = 0. Setting each factor equal to zero gives us two critical numbers: x = -7 and x = 5. Therefore, the critical numbers of the function f(x) = x³ + 3x² - 105x are -7 and 5.