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The sign of f′ describes monotonicity on an interval. Split the number line at all relevant critical inputs.
Worked example 1
For f(x)=(x−2)2, where is f decreasing?
f′(x)=2(x−2)
x<2⟹f′(x)<0; x>2⟹f′(x)>0
Requested interval/point: (−∞,2)
Answer:(−∞,2)
Worked example 2
For f(x)=(x−3)2, where is f increasing?
f′(x)=2(x−3)
x<3⟹f′(x)<0; x>3⟹f′(x)>0
Requested interval/point: (3,∞)
Answer:(3,∞)
Idea 2 of 3
First Derivative Test
A positive-to-negative derivative sign change gives a maximum; negative-to-positive gives a minimum.
Worked example 1
For f(x)=x3−12x, find the positive critical number.
f′(x)=3(x−2)(x+2)=0⟹x=±2
f′(x)>0 on (−∞,−2) and (2,∞); f′(x)<0 on (−2,2)
x
−2
2
f(x)
16
−16
requested value =2
Answer:2
Worked example 2
For f(x)=x3−27x, find the local minimum value.
f′(x)=3(x−3)(x+3)=0⟹x=±3
f′(x)>0 on (−∞,−3) and (3,∞); f′(x)<0 on (−3,3)
x
−3
3
f(x)
54
−54
requested value =−54
Answer:−54
Idea 3 of 3
Critical-point classification
A zero derivative alone is insufficient. Inspect the sign on both sides.
Worked example 1
For f(x)=(x−2)2, where is f decreasing?
f′(x)=2(x−2)
x<2⟹f′(x)<0; x>2⟹f′(x)>0
Requested interval/point: (−∞,2)
Answer:(−∞,2)
Worked example 2
For f(x)=(x−3)2, where is f increasing?
f′(x)=2(x−3)
x<3⟹f′(x)<0; x>3⟹f′(x)>0
Requested interval/point: (3,∞)
Answer:(3,∞)
Resource alignment: Larson & Edwards, Calculus of a Single Variable, Tenth Edition, §3.3,p. 177. Explanations, diagrams and numerical practice variants are original MathLab content aligned to this section.