Increasing and Decreasing Functions
A differentiable function increases on an interval where its derivative is non-negative and decreases where it is non-positive.
What is Increasing and Decreasing Functions?
A differentiable function increases on an interval where its derivative is non-negative and decreases where it is non-positive.
Key formula / rule: f'(x) > 0 on an interval implies f is strictly increasing there.
Key points
- Analyse the intervals of increase and decrease of a function.
- Prove that an equation has exactly one real root using monotonicity.
Common exam trap
Concluding monotonicity of a function on a union of intervals separated by a discontinuity.
Definitions
- Term
Increasing and Decreasing Functions
- Meaning
A differentiable function increases on an interval where its derivative is non-negative and decreases where it is non-positive.
- Term
Increasing and Decreasing Functions — explanation
- Meaning
The sign of the derivative is a complete description of direction of change, so monotonicity questions reduce to solving an inequality. Strict monotonicity needs the derivative to be zero only at isolated points.
Learning objectives
Analyse the intervals of increase and decrease of a function.
Prove that an equation has exactly one real root using monotonicity.
Formulae
- Key point
f'(x) > 0 on an interval implies f is strictly increasing there.
- Key point
The test applies on intervals, never at a single point.
- Key point
Monotonicity proves uniqueness of roots for equations.
Prerequisites
MAT-U2-LC-T2-S1-C1
Common mistakes
Concluding monotonicity of a function on a union of intervals separated by a discontinuity.
Confusing the sign of f' with the sign of f.
Keywords
Increasing
Decreasing
Functions
Practice preview
Find the values of 'a' for which the function f(x) = x^3 - 6x^2 + ax + 5 is strictly increasing on the interval [1, 2].…
hard
The function f(x) = x - log_e(1 + x) is strictly increasing on:…
hard
Find the interval where the function f(x) = x^2 - 4x + 6 is strictly increasing.…
easy