Critical Points and Local Extrema

Calculus 1 · Axiom Academy

LESSON Critical Points and Local Extrema Where a function can peak or bottom out, and how the sign of its derivative tells max from min. Slide along the curve and watch the tangent line . At a smooth peak or valley the tangent goes perfectly flat — its slope f'(c) hits 0 . At a sharp corner the tangent can't even be drawn — f'(c) is undefined . Those two events, and only those, are the critical points . f'(c) = 0 (the tangent is horizontal), or f'(c) is undefined (a corner, cusp, or vertical tangent). "Extreme" comes in two flavors. A point is a local extremum if it beats only its immediate neighbors ; it is an absolute extremum if it beats every point in the domain. Watch the window slide: inside it, each bump is a champion of its own little stretch, but only one point wins the whole graph. A local max at x=c means for all x near c . (Local min flips the inequality.) A function can have several. An absolute max at x=c means for all x in the domain — the single highest value anywhere. At most one each. To classify a critical point, read the sign of f' just to its left and just to its right. Where f' > 0 the curve climbs; where f' < 0 it falls. A switch from climbing to falling carves a peak ; falling to climbing carves a valley ; no switch means the point is neither. Local maximum: f' changes + → - Local minimum: f' changes - → + Neither: f' keeps the same sign (no change) 4. Putting It Together: a Sign Chart

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