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A critical point of f(x) is any x-value where f'(x) = 0 or f'(x) is undefined. Critical points are candidates for local maxima, local minima, or neither (saddle points / inflection points).
To find critical points: differentiate f(x), set f'(x) = 0, and solve. Also check where f'(x) does not exist (e.g., cusps, vertical tangents). Each solution is a critical number.
Once critical points are found, classify them using the second derivative test: f''(c) > 0 → local min; f''(c) < 0 → local max; f''(c) = 0 → inconclusive (use first derivative test).
Critical points are foundational to optimization, curve sketching, and understanding function behavior — they mark every location where the function's trend potentially changes direction.
f'(x) = 0 or f'(x) undefined → classify with f''(x)
Select a function
Second Derivative Test
f''(c) > 0 → Local Minimum (concave up)
f''(c) < 0 → Local Maximum (concave down)
f''(c) = 0 → Inconclusive — use 1st derivative test
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.