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

1 question

1 item

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

Solution: f'(x) = 6x^2 - 18x + 12. Setting f'(x) = 0 gives x = 1 and x = 2. f(1) = 5 is a local maximum.

Steps: 11

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.