For the function f(x) = x^3 - 6x^2 + 9x, find the critical points.

For the function f(x) = x^3 - 6x^2 + 9x, find the critical points.

Step 1: Start with the function f(x) = x^3 - 6x^2 + 9x.
Step 2: Find the derivative of the function, which is f'(x). The derivative tells us the rate of change of the function.
Step 3: Calculate f'(x) = 3x^2 - 12x + 9.
Step 4: Set the derivative equal to zero to find critical points: 3x^2 - 12x + 9 = 0.
Step 5: Simplify the equation by dividing everything by 3: x^2 - 4x + 3 = 0.
Step 6: Factor the quadratic equation: (x - 3)(x - 1) = 0.
Step 7: Solve for x by setting each factor equal to zero: x - 3 = 0 gives x = 3, and x - 1 = 0 gives x = 1.
Step 8: The critical points are x = 1 and x = 3.

Finding Critical Points – This involves taking the derivative of a function and setting it to zero to find points where the function's slope is zero, indicating potential local maxima, minima, or points of inflection.
Derivative Calculation – Understanding how to correctly differentiate polynomial functions is essential for finding critical points.
Factoring Quadratics – The ability to factor quadratic equations is necessary to solve for the values of x that make the derivative zero.