If f(x) = x^3 - 3x^2 + 4, find the critical points. (2022)

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Question: If f(x) = x^3 - 3x^2 + 4, find the critical points. (2022)

Options:

Correct Answer: 1, 2

Solution:

f\'(x) = 3x^2 - 6x. Setting f\'(x) = 0 gives x(3x - 6) = 0, so x = 0 or x = 2.

If f(x) = x^3 - 3x^2 + 4, find the critical points. (2022)

If f(x) = x^3 - 3x^2 + 4, find the critical points. (2022)

Step 1: Start with the function f(x) = x^3 - 3x^2 + 4.
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 the derivative: f'(x) = 3x^2 - 6x.
Step 4: Set the derivative equal to zero to find critical points: 3x^2 - 6x = 0.
Step 5: Factor the equation: 3x(x - 2) = 0.
Step 6: Solve for x by setting each factor equal to zero: 3x = 0 or x - 2 = 0.
Step 7: From 3x = 0, we get x = 0. From x - 2 = 0, we get x = 2.
Step 8: The critical points are x = 0 and x = 2.

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 differentiate polynomial functions correctly is essential for finding critical points.
Factoring Polynomials – The ability to factor expressions to solve for roots is crucial in determining the critical points from the derivative.