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First & Second Derivative Test
AP Calculus AB · Unit 5 Analytical Applications
First Derivative Test
At critical point (f′ = 0):
• f′ changes + to − →
local max
• f′ changes − to + →
local min
• f′ same sign → neither (no extremum)
Second Derivative Test
At critical point c (f′(c) = 0):
• f″(c) > 0 →
local min
(concave up)
• f″(c) < 0 →
local max
(concave down)
• f″(c) = 0 →
inconclusive
Tags
derivative test
extrema
critical points
max/min
FUNCTION
f(x) = x³ − 3x + 1
f(x) = x⁴ − 4x²
f(x) = x·e⁻ˣ
f(x) = sin(x) + x/2