Starting Point: Critical Points
Both tests start from critical points — values of x where f'(x) = 0 or f'(x) is undefined. You cannot test a point that is not critical.
The First Derivative Test
Build a sign chart for f'(x) across the critical points. The sign pattern tells you the classification:
f' changes + to − at c → local maximum
f' changes − to + at c → local minimum
f' does not change sign at c → neither (saddle point)
The Second Derivative Test
Faster when f''(c) is easy to compute. Only works at stationary points where f'(c) = 0:
f''(c) < 0 → local maximum (concave down at c)
f''(c) > 0 → local minimum (concave up at c)
f''(c) = 0 → inconclusive (must use first test)
Example 1 — f(x) = x³ − 6x² + 9x
📋 Classify all critical points
f'(x)= 3x²−12x+9 = 3(x−1)(x−3). Critical pts: x=1, x=3
f''(x)= 6x−12. f''(1) = −6 < 0 → local max. f''(3) = 6 > 0 → local min
Valuesf(1) = 1−6+9 = 4 (local max). f(3) = 27−54+27 = 0 (local min)
Example 2 — First Derivative Test Sign Chart
📋 f(x) = x⁴ − 4x³
f'(x)= 4x³−12x² = 4x²(x−3). Critical pts: x=0, x=3
Sign chartx<0: f'= 4(+)(−) < 0. 0<x<3: f'= 4(+)(−) < 0. x>3: f'= 4(+)(+) > 0
x=0f' goes − to − → no sign change → neither (saddle point)
x=3f' goes − to + → local minimum. f(3) = 81−108 = −27
Example 3 — Second Derivative Test Fails (f''=0)
📋 f(x) = x⁴ at x=0
f'(0)=0Critical point at x=0 ✓
f''(0)=0Inconclusive! Must use first derivative test
Sign chartf'= 4x³: negative for x<0, positive for x>0 → − to + → local minimum
Example 4 — Trig Function
📋 f(x) = sin(x) + cos(x) on [0, 2π]
f'(x)=0cos(x)−sin(x)=0 → tan(x)=1 → x=π/4, 5π/4
f''(x)= −sin(x)−cos(x). f''(π/4) = −√2 < 0 → local max. f''(5π/4) = √2 > 0 → local min
Example 5 — Absolute Extrema on Closed Interval
📋 f(x) = x³−3x on [−2,2]
Critical ptsf'=3x²−3=0 → x=±1
Evaluate allf(−2)=−2, f(−1)=2, f(1)=−2, f(2)=2
Absolute maxf=2 at x=−1 and x=2
Absolute minf=−2 at x=1 and x=−2
Which Test to Use When?
Decision Guide
Use the second derivative test when f'' is easy to compute and you expect f''(c) ≠ 0. Use the first derivative test when f''(c) = 0 (second test fails), when f' is hard to differentiate again, or when you need the complete sign pattern of f' for graphing.
Frequently Asked Questions
How do I build a sign chart for f'?
Mark all critical points on a number line. They divide the real line into intervals. Pick one test value inside each interval, plug it into f'(x), and record the sign (+ or −). The pattern tells you where f is increasing (+) or decreasing (−), and whether each critical point is a max, min, or neither.
What is the difference between local and absolute extrema?
A local maximum is just the highest point in some neighbourhood. The absolute maximum is the highest point on the entire domain. On a closed interval, the absolute extrema occur at critical points or at the endpoints. Always evaluate f at both critical points and endpoints to find absolute extrema.