For the function f(x) = 2x^3 - 9x^2 + 12x, find the local maxima.

For the function f(x) = 2x^3 - 9x^2 + 12x, find the local maxima.

Step 1: Start with the function f(x) = 2x^3 - 9x^2 + 12x.
Step 2: Find the derivative of the function, which is f'(x). The derivative tells us the slope of the function.
Step 3: Calculate the derivative: f'(x) = 6x^2 - 18x + 12.
Step 4: Set the derivative equal to zero to find critical points: 6x^2 - 18x + 12 = 0.
Step 5: Simplify the equation by dividing everything by 6: x^2 - 3x + 2 = 0.
Step 6: Factor the quadratic equation: (x - 1)(x - 2) = 0.
Step 7: Solve for x: This gives us x = 1 and x = 2 as critical points.
Step 8: To determine if these points are local maxima, we need to evaluate the function at these points: f(1) and f(2).
Step 9: Calculate f(1): f(1) = 2(1)^3 - 9(1)^2 + 12(1) = 2 - 9 + 12 = 5.
Step 10: Calculate f(2): f(2) = 2(2)^3 - 9(2)^2 + 12(2) = 16 - 36 + 24 = 4.
Step 11: Compare the values: f(1) = 5 and f(2) = 4. Since f(1) is greater than f(2), x = 1 is a local maximum.

Finding Local Maxima – This involves taking the derivative of a function, setting it to zero to find critical points, and using the second derivative test or evaluating the function at those points to determine if they are local maxima or minima.
Critical Points – Critical points are found where the first derivative is zero or undefined, which are candidates for local maxima or minima.
Second Derivative Test – This test helps confirm whether a critical point is a local maximum, local minimum, or neither by evaluating the second derivative at that point.