x^3-6x^2+8 [1,6] local minimum or maximum classify each value you find if any as a minimum or maximum



To determine whether there are local minima or maxima in the function \( f(x) = x^3 - 6x^2 + 8 \) on the interval \([1, 6]\), we will follow these steps:

1. **Find the first derivative** \( f'(x) \) and set it to zero to locate critical points.
2. **Find the second derivative** \( f''(x) \) to classify each critical point as a minimum, maximum, or neither.
3. **Evaluate** the function at each critical point and at the endpoints of the interval to determine the minimum and maximum values within \([1, 6]\).

### Step 1: First Derivative
The first derivative of \( f(x) = x^3 - 6x^2 + 8 \) is:
\[
f'(x) = 3x^2 - 12x
\]
We can factor this as:
\[
f'(x) = 3x(x - 4)
\]
Setting \( f'(x) = 0 \):
\[
3x(x - 4) = 0
\]
This gives \( x = 0 \) and \( x = 4 \) as critical points. However, since we are only interested in the interval \([

x^3 - 6x^2 + 8 [1,6] local minimum or maximum classify each value you find if any as a minimum or maximum

Explore the step-by-step solution for finding and classifying the local minima and maxima of the function f(x) = x^3 - 6x^2 + 8 on the interval [1,6]. This problem involves calculating derivatives, identifying critical points, and applying the First and Second Derivative Tests, ideal for students in Grades 11-12.

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