Question

Let f be the function given by f(x)=x³-6x². What is the local extrema?

Answer 1

To identify the local extrema of the function f(x) = x³ - 6x², you find critical points by setting the first derivative f'(x) = 3x² - 12x to zero. Critical points at x = 0 and x = 4 then need to be tested to determine if they are local maxima or minima.

Step-by-step explanation:

To find the local extrema of the function f(x) = x³ - 6x², you need to follow the steps for finding the critical points and then determine whether these points are local maxima, minima, or neither.

1. First, find the first derivative of f(x): f'(x) = 3x² - 12x.
2. Set the first derivative equal to zero to find critical points: 3x² - 12x = 0.
3. Factor out the common term: x(3x - 12) = 0, which gives us x = 0 and x = 4 as critical points.
4. Determine the nature of each critical point by using the second derivative test or by analyzing the sign changes of the first derivative around the critical points.

After examining the behavior of the function around the critical points, we can classify them as local maxima, minima, or neither to identify the local extrema of the function f(x).