What Are Critical Points?
A critical point of a function f(x) is a point where the derivative f'(x) equals zero or is undefined. Critical points are essential in calculus for finding local maxima, local minima, and inflection points of functions.
A point x = c is critical if f'(c) = 0 or f'(c) is undefined.
For polynomials: Set f'(x) = 0 and solve for x.
Finding Critical Points
- Take the derivative f'(x)
- Set f'(x) = 0
- Solve for x
- Use the second derivative test to classify each point
Classifying Critical Points
Second Derivative Test: If f''(c) > 0, the point is a local minimum. If f''(c) < 0, it is a local maximum. If f''(c) = 0, the test is inconclusive.
Polynomial Critical Points
For a polynomial f(x) = ax³ + bx² + cx + d, the derivative is f'(x) = 3ax² + 2bx + c. Setting this equal to zero and using the quadratic formula gives the critical points.
How to Use
Enter the coefficients of your polynomial function. The calculator finds the derivative, solves for critical points, and classifies them as maxima or minima.
Applications
Critical points are used in optimization problems (maximizing profit, minimizing cost), curve sketching, physics (equilibrium points), and engineering (stress analysis).