Step 7: Solve for x: x = 1 and x = 2 are the critical points.
Step 8: To find local maxima, we need to check the second derivative or test the values around the critical points.
Step 9: Calculate the second derivative, f''(x) = 12x - 18.
Step 10: Evaluate f''(2): f''(2) = 12(2) - 18 = 6, which is positive, indicating a local minimum at x = 2.
Step 11: Evaluate f''(1): f''(1) = 12(1) - 18 = -6, which is negative, indicating a local maximum at x = 1.
Step 12: Find the value of the function at the local maximum: f(1) = 2(1)^3 - 9(1)^2 + 12(1) = 2 - 9 + 12 = 5.
Finding Local Maxima – The process of determining points where a function reaches a local maximum by using the first derivative test.
Critical Points – Identifying points where the first derivative is zero or undefined to find potential local maxima or minima.
Second Derivative Test – A method to confirm whether a critical point is a local maximum, local minimum, or neither by evaluating the second derivative.
