Find increasing and decreasing intervals for common function types
Select Function Type
f(x)=ax+b
f(x)=x +
f(x)=ax^2+bx+c
f(x)=x^2 +x +
f(x)=ax^3+bx^2
f(x)=x^3 +x^2
f(x)=1/(ax+b)
f(x)=1/(x +)
Result
Derivation
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.
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