Step 7: Solve for x: This gives us x = 1 and x = 2.
Step 8: To find out if these points are local minima or maxima, we can use the second derivative test or check the values of f(x) around these points.
Step 9: Calculate f(2) to find the function value at x = 2: f(2) = 2(2)^3 - 9(2)^2 + 12(2) = 0.
Step 10: Since we are looking for a local minimum, we check the second derivative or the behavior of f(x) around x = 2.
Finding Local Minima – The question tests the ability to find local minima of a function using calculus, specifically by finding critical points through the first derivative.
Critical Points – Identifying where the first derivative equals zero to find potential local extrema.
Second Derivative Test – Understanding that further analysis (like the second derivative test) is needed to confirm whether the critical point is a minimum or maximum.
