Determine the local maxima of f(x) = -x^3 + 3x^2 + 1. (2021)

Determine the local maxima of f(x) = -x^3 + 3x^2 + 1. (2021)

Step 1: Write down the function f(x) = -x^3 + 3x^2 + 1.
Step 2: Find the derivative of the function, which is f'(x) = -3x^2 + 6x.
Step 3: Set the derivative equal to zero to find critical points: -3x^2 + 6x = 0.
Step 4: Factor the equation: -3x(x - 2) = 0.
Step 5: Solve for x: This gives us x = 0 and x = 2.
Step 6: To determine if these points are local maxima, we need to evaluate the function at these points.
Step 7: Calculate f(0): f(0) = -0^3 + 3(0)^2 + 1 = 1.
Step 8: Calculate f(2): f(2) = -2^3 + 3(2)^2 + 1 = -8 + 12 + 1 = 5.
Step 9: Compare the values: f(0) = 1 and f(2) = 5. Since 5 is greater than 1, x = 2 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 to determine if these points are local maxima or minima.
Critical Points – Critical points are 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 sign of the second derivative at that point.