Critical Points

Business Calculus · Axiom Academy

Finding candidates for maxima and minima A number c in the domain of f is a critical point (or critical number) if either: Critical points are the only places where a local maximum or minimum can occur: The function levels off momentarily. Think of the top of a hill or bottom of a valley. The function has a corner point where the slope changes abruptly. Find where f'(x) is undefined (but f(x) is defined) List all critical points from steps 2 and 3 Let f(x) = x^3 - 6x^2 + 9x + 1 Critical Points: x = 1 and x = 3 Since f'(x) is a polynomial, it's defined everywhere, so we only have critical points where f'(x) = 0. At each critical point, observe what happens to the function: The function has a local maximum . The curve goes up, levels off, then goes down. The function has a local minimum . The curve goes down, levels off, then goes up. Not every critical point is a maximum or minimum! We need additional tests (coming next) to determine what type of critical point we have. Example with Undefined Derivative This absolute value function has a corner at x = 2. We can write this as a piecewise function: At x = 2: f'(x) does not exist The left derivative is -1 and the right derivative is +1. Since they don't match, the derivative doesn't exist at x = 2. This is a critical point! Even though f'(2) doesn't exist, the function f(x) = |x - 2| clearly has a minimum at x = 2. Finding Critical Points in Business Step 3: Check for undefined derivatives

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