Find the value of x where the function f(x) = x^3 - 6x^2 + 9x has a local minimu

Find the value of x where the function f(x) = x^3 - 6x^2 + 9x has a local minimum. (2020)

Find the value of x where the function f(x) = x^3 - 6x^2 + 9x has a local minimum. (2020)

Step 1: Write down the function f(x) = x^3 - 6x^2 + 9x.
Step 2: Find the derivative of the function, which is f'(x) = 3x^2 - 12x + 9.
Step 3: Set the derivative equal to zero to find critical points: 3x^2 - 12x + 9 = 0.
Step 4: Simplify the equation by dividing everything by 3: x^2 - 4x + 3 = 0.
Step 5: Factor the quadratic equation: (x - 1)(x - 3) = 0.
Step 6: Solve for x to find critical points: x = 1 and x = 3.
Step 7: To determine if these points are local minima or maxima, test a value between them, such as x = 2.
Step 8: Calculate f'(2): f'(2) = 3(2)^2 - 12(2) + 9 = 3(4) - 24 + 9 = 12 - 24 + 9 = -3.
Step 9: Since f'(2) is negative, it indicates that the function is decreasing before x = 2.
Step 10: Now test a value greater than 3, such as x = 4: f'(4) = 3(4)^2 - 12(4) + 9 = 48 - 48 + 9 = 9.
Step 11: Since f'(4) is positive, it indicates that the function is increasing after x = 2.
Step 12: Therefore, x = 2 is a local minimum.

Finding Local Minima – The question tests the ability to find local minima of a polynomial function using calculus, specifically by finding critical points and using the first derivative test.
Critical Points – Identifying critical points by setting the first derivative to zero and solving for x.
Second Derivative Test – Understanding how to confirm whether a critical point is a local minimum or maximum, although not explicitly required in the solution.