Question

Find all value(s) of x where f(x) = 2x³ - 3x² - 12x has a local minimum.

a) x = -2
b) x = 1
c) x = 2
d) x = -1

Answer 1

To find the values of x where f(x) = 2x³ - 3x² - 12x has a local minimum, we need to find the critical points of the function. The local minimum occurs at a point where the derivative changes sign from negative to positive. The values of x where f(x) has a local minimum are x = -2, 1, and 2.

Step-by-step explanation:

To find the values of x where f(x) = 2x³ - 3x² - 12x has a local minimum, we need to find the critical points of the function. The local minimum occurs at a point where the derivative changes sign from negative to positive. To find the derivative, we take the derivative of f(x) with respect to x and set it equal to zero:

f'(x) = 6x² - 6x - 12 = 0

Now we can solve this quadratic equation to find the values of x:

6x² - 6x - 12 = 0

Using factoring or the quadratic formula, we find that x = -2, 1, and 2. Therefore, the values of x where f(x) has a local minimum are x = -2, 1, and 2.