Monotonicity of Functions
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Direct answer
A differentiable function f is strictly increasing on an interval where f'(x) > 0 for every x, and strictly decreasing where f'(x) < 0 for every x; if f'(x) >= 0 throughout with equality only at isolated points, f is still increasing — x^3 at 0 is the canonical example. Monotonicity is therefore read from the sign chart of f' across the interval, not from the zeros of f'.
What you must remember
- Core test: f' > 0 gives strictly increasing, f' < 0 gives strictly decreasing; f' >= 0 with isolated zeros still gives increasing on the interval.
- Critical points: where f' = 0 or f' does not exist; for continuous functions the sign of f' can change only across such points, so they fence the increasing and decreasing intervals.
- Sign chart method: factorise f'(x) completely, place all factors' roots on a line, and read the sign of the product in each interval — wavy-curve logic applied to derivatives.
- Squares and even powers do not change sign: f'(x) = (x - a)(x - b)^2/(x^2 + 1) changes sign only across x = a, since (x - b)^2 and the denominator stay positive.
- Composite rules: increasing of increasing is increasing; increasing of decreasing is decreasing; the inverse of a strictly monotonic function is strictly monotonic on the image.
- Local extremum link: f' changing from + to - at c gives a local maximum, from - to + a local minimum, and no sign change neither.
- Injectivity: a strictly monotonic function on an interval is one-one — the standard route to proving that an equation has at most one solution.
Common confusion
The damaging error is treating f'(c) = 0 as evidence of an extremum, or f' touching zero as evidence of non-monotonicity. x^3 has f'(0) = 0 yet is strictly increasing on the whole line; x^4 has f'(0) = 0 and a genuine minimum — the sign chart decides, not the zero. Monotonicity is also relative to an interval: never state "f is increasing" without naming one.
Exam-focused takeaway
JEE Main asks for the intervals of increase or decrease of moderately complicated functions — factorise the derivative, deploy the sign chart, report intervals as numerical answers. JEE Advanced weaponises the same idea: proving inequalities by studying g(x) = f(x) - h(x), showing g' > 0 and comparing g at an endpoint; counting roots of equations via monotonicity; and chaining composite monotonicity statements. Wherever an inequality must be proved "for all x > 0", expect monotonicity to be the engine, not brute-force algebra.
Frequently asked questions
Does f'(c) = 0 mean c is a local extremum?
Not necessarily. For x^3 at 0 the derivative vanishes yet the function is strictly increasing; check whether f' actually changes sign at c.
Can a strictly increasing function have f' = 0 somewhere?
Yes, provided the zeros are isolated — x^3 is strictly increasing on all real numbers despite f'(0) = 0.
How do I find the intervals where f increases?
Solve f'(x) > 0 after factorising the derivative; the roots of the factors split the line into intervals, and the sign chart labels each as increasing or decreasing.
How is monotonicity used to prove inequalities?
Define g = left side minus right side, show g' keeps one sign on the interval, and compare g at the endpoint; monotonicity then transfers the endpoint inequality to the whole interval.
Does monotonicity imply the function is one-one?
Yes, on the interval where it is strictly monotonic — the standard proof of injectivity.