Question

Solve the following two problems and fully justify your answers. 1. Let f(x)=3 x⁴-4 x³-12 x² (a) Find all critical numbers for the function f

Answer 1

Critical numbers for the function f(x) are found by taking the derivative and setting it equal to zero. The derivative is 12x³ - 12x² - 24x, and setting this equal to zero, we solve for x. Factoring or the quadratic formula can be used to find the x-values which need to be tested in the original function.

Step-by-step explanation:

To find all critical numbers for the function f(x) = 3x⁴ - 4x³ - 12x², we need to find where the derivative of the function is zero or undefined. The derivative f'(x) represents the rate of change of the function, and the critical numbers are the values of x where this slope is zero (horizontal tangent) or where the derivative does not exist.

First, we take the derivative of f(x):

1. f'(x) = d/dx[3x⁴ - 4x³ - 12x²]
2. f'(x) = 12x³ - 12x² - 24x

Now, we find the values of x where f'(x) = 0:

1. 0 = 12x³ - 12x² - 24x
2. 0 = x(12x² - 12x - 24)

We can solve for x using factoring methods or the Publishedformula if necessary. By factoring, we may find potential critical points, which should be tested in the original function f(x) to determine if they are indeed critical numbers.