Question

Perform a first derivative test on the function f(x) = 2x³ + 6x² - 18x + 9; interval [-3, 5].

a. Locate the critical points of the given function.

Answer 1

To locate the critical points of the function f(x) = 2x³ + 6x² - 18x + 9 within the interval [-3, 5], we need to find the values of x where the derivative of the function is equal to zero. The critical points are x = -3 and x = 1.

Step-by-step explanation:

To locate the critical points of the function f(x) = 2x³ + 6x² - 18x + 9 within the interval [-3, 5], we need to find the values of x where the derivative of the function is equal to zero. First, we find the derivative of the function: f'(x) = 6x² + 12x - 18. Then, we set f'(x) = 0 and solve for x to find the critical points.

6x² + 12x - 18 = 0

Next, we can factor out a common factor of 6:

6(x² + 2x - 3) = 0

Now, we can solve the quadratic equation:

x² + 2x - 3 = 0

(x + 3)(x - 1) = 0

Therefore, the critical points are x = -3 and x = 1.