Step 6: Solve for x: This gives us the critical points x = 1 and x = 3.
Step 7: To find if these points are local minima or maxima, we can use the second derivative test.
Step 8: Find the second derivative: f''(x) = 6x - 12.
Step 9: Evaluate the second derivative at the critical points: f''(1) = 6(1) - 12 = -6 (which is less than 0, so x = 1 is a local maximum) and f''(3) = 6(3) - 12 = 6 (which is greater than 0, so x = 3 is a local minimum).
Step 10: To find the value of the local minimum, calculate f(3): f(3) = 3^3 - 6(3^2) + 9(3) = 27 - 54 + 27 = 0.
Step 11: Therefore, the local minimum occurs at x = 3 with a value of f(3) = 0.
Finding Local Minima – The process of determining points where a function reaches a local minimum by analyzing its first derivative.
Critical Points – Points where the first derivative is zero or undefined, which are candidates for local extrema.
Second Derivative Test – A method to confirm whether a critical point is a local minimum, local maximum, or neither by evaluating the second derivative.
