👉 The accompanying math involves the application of calculus to analyze the behavior of a function f(x) = x^3 - 6x^2 + 9x - 4, specifically to find its critical points and determine the nature of these points (local maxima, minima, or saddle points). By taking the first derivative, f'(x) = 3x^2 - 12x + 9, we identify the critical points where f'(x) = 0. Factoring this gives (3x - 3)(x - 3) = 0, leading to x = 1 and x = 3 as critical points. Using the second derivative test, f''(x) = 6x - 12, we evaluate it at these points: f''(1) = -6 (negative), indicating a local maximum at x = 1, and f''(3) = 6 (positive), indicating a local minimum at x = 3. This analysis provides insights into the function's concavity and extreme values, crucial for understanding its graph and behavior.
