How to solve the critical point of a function Local minimum local maximum

To find the critical points, local minima, and local maxima of a function f(x): Step 1: Find the first derivative of the function, f'(x). f'(x) = (d)/(dx) f(x) Step 2: Find the critical points by setting the first derivative equal to zero and solving for x, or by finding where the first derivative is undefined. f'(x) = 0 or f'(x) is undefined The x-values obtained are the critical points. Step 3: Use either the First Derivative Test or the Second Derivative Test to classify these critical points as local minima, local maxima, or neither. • First Derivative Test: Choose test values in intervals around each critical point. If f'(x) changes from positive to negative at a critical point, it's a local maximum*. If f'(x) changes from negative to positive at a critical point, it's a local minimum*. If f'(x) does not change sign, it's neither (often an inflection point). • Second Derivative Test: Find the second derivative of the function, f''(x). Substitute each critical point x_c into f''(x). If f''(x_c) > 0, the critical point is a local minimum*. If f''(x_c) < 0, the critical point is a local maximum*. If f''(x_c) = 0, the test is inconclusive, and you should use the First Derivative Test. That's 2 down. 3 left today — send the next one.