Find Critical Points 815449

1. **State the problem:** We need to find the critical points of a function. Critical points occur where the derivative of the function is zero or undefined. 2. **Formula and rules:** The critical points are found by solving $$f'(x) = 0$$ or where $$f'(x)$$ does not exist. 3. **Steps to find critical points:** - Find the derivative $$f'(x)$$ of the given function $$f(x)$$. - Solve the equation $$f'(x) = 0$$ to find potential critical points. - Check where $$f'(x)$$ is undefined. 4. **Example:** Suppose $$f(x) = x^3 - 3x^2 + 4$$. - Compute derivative: $$f'(x) = 3x^2 - 6x$$. - Set derivative to zero: $$3x^2 - 6x = 0$$. - Factor: $$3x(x - 2) = 0$$. - Solve: $$x = 0$$ or $$x = 2$$. - Check if derivative undefined anywhere: derivative is a polynomial, so defined everywhere. 5. **Conclusion:** The critical points are at $$x = 0$$ and $$x = 2$$. You can apply this method to any differentiable function to find its critical points.