Final answer:
To find the critical points of the function 4x^3 + 24x^2, we derive the function to get 12x^2 + 48x and factor it to 12x(x + 4). Setting this equal to zero gives the critical points x = 0 and x = -4.
Step-by-step explanation:
The student is asking to find the critical points of the function 4x^3 + 24x^2. To find the critical points, we need to take the derivative of the function and set it equal to zero. The derivative of this function is 12x^2 + 48x. We can factor out a 12x to simplify the equation to 12x(x + 4). Setting this equation to zero, we get the critical points at x = 0 and x = -4. These are the values of x where the slope of the tangent to the curve is zero or where the derivative does not exist.