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Home›Function Tools›Monotonicity Checker

Monotonicity Checker

Find increasing and decreasing intervals for common function types

Select Function Type

Linear ax+b Quadratic ax^2+bx+c Cubic ax^3+bx^2 Rational 1/(ax+b)

f(x)=ax+b

f(x)=x +

Monotonicity Rules

Linear: f(x)=a, a>0 increasing, a<0 decreasing

Quadratic: decreasing then increasing (a>0)

Critical points divide monotonic intervals

Check f(x) sign: >0 increasing, <0 decreasing

Monotonicity describes whether a function is consistently increasing or decreasing on an interval. The derivative sign determines the behavior: positive derivative means increasing, negative means decreasing. Critical points separate intervals of different monotonicity.

⚠ A function can be monotonic on a subset of its domain. Non-monotonic functions change direction at critical points.

What Is Monotonicity?

A function is monotonic increasing if larger inputs give larger outputs. The derivative f(x) measures the instantaneous rate of change. Positive sign = increasing, negative = decreasing. Critical points where f(x)=0 or undefined are boundaries between intervals.

Linear

f(x)=a. If a>0: always increasing. a<0: always decreasing. a=0: constant (monotonic).

Quadratic

f(x)=2ax+b. Split at vertex x=-b/(2a). One side increasing, other decreasing.

Cubic

f(x)=3ax^2+2bx. Quadratic derivative. Can have 2 critical points dividing into 3 intervals.

Rational 1/(ax+b)

f(x)=-a/(ax+b)^2. Sign opposite of a. Vertical asymptote divides domain into 2 intervals, each monotonic.

Teaching Example: f(x)=x^2-4x+3. f(x)=2x-4=0 at x=2. x<2: f negative -> decreasing. x>2: f positive -> increasing. Vertex (2,-1) is minimum.

Applications

Graphing Optimization Calculus Inverse Functions Algorithms

Frequently Asked Questions

What is monotonicity?▼

Function behavior: increasing (f>0) or decreasing (f<0) on intervals. Critical points divide the domain.

How to find intervals?▼

Find f(x), find critical points, test sign in each interval between critical points. >0 increasing, <0 decreasing.

Linear monotonicity?▼

f(x)=ax+b. Derivative=a. a>0 always increasing. a<0 always decreasing. a=0 constant.

Why does rational split?▼

Vertical asymptote divides domain. Each side is monotonic (always increasing or always decreasing depending on sign).

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