For the function f(x) = 3x^3 - 12x^2 + 9x, the number of local maxima and minima

For the function f(x) = 3x^3 - 12x^2 + 9x, the number of local maxima and minima is:

For the function f(x) = 3x^3 - 12x^2 + 9x, the number of local maxima and minima is:

Step 1: Start with the function f(x) = 3x^3 - 12x^2 + 9x.
Step 2: Find the first derivative f'(x) to determine critical points. The first derivative is f'(x) = 9x^2 - 24x + 9.
Step 3: Set the first derivative equal to zero to find critical points: 9x^2 - 24x + 9 = 0.
Step 4: Solve the quadratic equation for x using the quadratic formula or factoring. This will give you two critical points.
Step 5: Find the second derivative f''(x) to determine the nature of the critical points. The second derivative is f''(x) = 18x - 24.
Step 6: Evaluate the second derivative at each critical point. If f''(x) > 0, it's a local minimum; if f''(x) < 0, it's a local maximum.
Step 7: Conclude that there is one local maximum and one local minimum based on the second derivative test.

Critical Points – Finding where the first derivative is zero or undefined to determine potential local maxima and minima.
Second Derivative Test – Using the second derivative to classify critical points as local maxima, minima, or points of inflection.
Polynomial Functions – Understanding the behavior of cubic functions and their derivatives.