For the function f(x) = 2x^3 - 9x^2 + 12x, find the critical points. (2022)

For the function f(x) = 2x^3 - 9x^2 + 12x, find the critical points. (2022)

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).
Step 3: Use the power rule to differentiate: 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 by setting each factor to zero: x - 1 = 0 or x - 2 = 0.
Step 8: This gives us the solutions: x = 1 and x = 2.
Step 9: The critical points are x = 1 and x = 2.

Finding Critical Points – This involves taking the derivative of the function and setting it to zero to find points where the function's slope is zero.
Derivative Calculation – Understanding how to correctly compute the derivative of a polynomial function.
Quadratic Equation Solutions – Solving the resulting quadratic equation to find the values of x that yield critical points.