Rule

• Let $f$ be differentiable on the open interval $]a,b[$.
• If $f^{\mathrm{\prime }}(x)>0$ for every $x\in ]a,b[$, then $f$ is increasing on $]a,b[$.
• If $f^{\mathrm{\prime }}(x)<0$ for every $x\in ]a,b[$, then $f$ is decreasing on $]a,b[$.

• Find the domain of $f$ and check where it is continuous and differentiable.
• Find $f^{\mathrm{\prime }}(x)$.
• Solve $f^{\mathrm{\prime }}(x)=0$ and find where $f^{\mathrm{\prime }}(x)$ does not exist; these values separate the intervals.
• Make a table of signs of $f^{\mathrm{\prime }}(x)$ on each interval (test one value in each).
• State the intervals of increase and decrease.

Examples

✍️ Worked examples

• Example 1 Find the intervals of increase and decrease of $f(x)=x^3-3x+2$.
• Solution:
• $f$ is a polynomial, so it is continuous and differentiable on $\mathbb{R}$.
• $f^{\mathrm{\prime }}(x)=3x^2-3=3(x-1)(x+1)$.
• $f^{\mathrm{\prime }}(x)=0$ at $x=-1$ and $x=1$.
• Sign of f′ positive on ]−∞,−1[, negative on ]−1,1[, positive on ]1,∞[.
• So $f$ is increasing on $]-\mathrm{\infty },-1[$ and on $]1,\mathrm{\infty }[$, and decreasing on $]-1,1[$.
• Example 2 Find the intervals of monotony of $f(x)=x+2\sin x$ for $0<x<2\pi$.
• Solution:
• $f^{\mathrm{\prime }}(x)=1+2\cos x$.
• $f^{\mathrm{\prime }}(x)=0$ gives $\cos x=-{\displaystyle \frac{1}{2}}$, so $x={\displaystyle \frac{2\pi }{3}}$ or $x={\displaystyle \frac{4\pi }{3}}$.
• Test points at x=π2, f′=1>0; at x=π, f′=−1<0; at x=3π2, f′=1>0.
• Increasing on $]0,{\displaystyle \frac{2\pi }{3}}[$ and $]{\displaystyle \frac{4\pi }{3}},2\pi [$; decreasing on $]{\displaystyle \frac{2\pi }{3}},{\displaystyle \frac{4\pi }{3}}[$.
• Example 3 Find the intervals of monotony of $h(x)=2\ln x-x^2$.
• Solution:
• The domain is $]0,\mathrm{\infty }[$.
• $h^{\mathrm{\prime }}(x)={\displaystyle \frac{2}{x}}-2x={\displaystyle \frac{2(1-x^2)}{x}}$.
• $h^{\mathrm{\prime }}(x)=0$ gives $x=1$ (since $x>0$).
• For $0<x<1$, $h^{\mathrm{\prime }}>0$; for $x>1$, $h^{\mathrm{\prime }}<0$.
• $h$ is increasing on $]0,1[$ and decreasing on $]1,\mathrm{\infty }[$.
• Example 4 Find the intervals of monotony of $f(x)={\displaystyle \frac{x}{x^2+1}}$.
• Solution:
• $f^{\mathrm{\prime }}(x)={\displaystyle \frac{(x^2+1)-2x^2}{(x^2+1{)}^2}}={\displaystyle \frac{1-x^2}{(x^2+1{)}^2}}$.
• The denominator is always positive, so the sign depends on $1-x^2$.
• $f^{\mathrm{\prime }}(x)>0$ on $]-1,1[$ and $f^{\mathrm{\prime }}(x)<0$ on $]-\mathrm{\infty },-1[$ and $]1,\mathrm{\infty }[$.
• $f$ increases on $]-1,1[$ and decreases on the other two intervals.
• Example 5 Prove that $f(x)=\tan x-x$ is increasing on $]0,{\displaystyle \frac{\pi }{2}}[$.
• Solution:
• $f^{\mathrm{\prime }}(x)={\sec }^{2}x-1={\tan }^{2}x$.
• On $]0,{\displaystyle \frac{\pi }{2}}[$, $\tan x>0$, so $f^{\mathrm{\prime }}(x)>0$.
• Hence $f$ is increasing there.

Notes

• The values that separate increasing from decreasing are those where $f^{\mathrm{\prime }}(x)=0$ or $f^{\mathrm{\prime }}(x)$ does not exist.
• A function can be strictly increasing even if $f^{\mathrm{\prime }}$ is zero at one isolated point, such as $f(x)=x^3$ at $x=0$.
• If $f^{\mathrm{\prime }}<h^{\mathrm{\prime }}$ everywhere, then $z=f-h$ has $z^{\mathrm{\prime }}=f^{\mathrm{\prime }}-h^{\mathrm{\prime }}<0$ everywhere, so $z$ is decreasing on $\mathbb{R}$.
• Graphing software confirms the table of signs the curve rises where the table says "+" and falls where it says "−".

Definition

• Let $f$ be defined on $[a,b]$ and $c\in [a,b]$.
• $f(c)$ is the absolute minimum if $f(c)\le f(x)$ for all $x\in [a,b]$.
• $f(c)$ is the absolute maximum if $f(c)\ge f(x)$ for all $x\in [a,b]$.
• Together they are the absolute extrema (extreme values). They may occur at an inner point or at an end point.

• If $f$ is continuous on $[a,b]$, it always has both an absolute maximum and an absolute minimum there.
• For an open or unbounded interval, or a discontinuous function, they may not exist.

• Find $f(a)$ and $f(b)$.
• Find the critical points in $]a,b[$ and the value of $f$ at each.
• Compare all these values the largest is the absolute maximum and the smallest is the absolute minimum.

✍️ Worked examples

• Example 11 Find the absolute extrema of $f(x)=x^3-12x+12$ on $[-3,3]$.
• Solution:
• $f(-3)=-27+36+12=21$ and $f(3)=27-36+12=3$.
• $f^{\mathrm{\prime }}(x)=3x^2-12=3(x-2)(x+2)$, so the critical points are $x=\pm 2$, both inside the interval.
• $f(2)=8-24+12=-4$ and $f(-2)=-8+24+12=28$.
• Compare 21,3,−4,28 absolute maximum =28 at x=−2, absolute minimum =−4 at x=2.
• Example 12 Find the absolute extrema of $f(x)=x{e}^{-x}$ on $[0,2]$.
• Solution:
• $f^{\mathrm{\prime }}(x)=(1-x)e^{-x}$, critical point $x=1$.
• $f(0)=0$, $f(1)={\displaystyle \frac{1}{e}}\approx 0.368$, $f(2)={\displaystyle \frac{2}{e^2}}\approx 0.271$.
• Absolute maximum ${\displaystyle \frac{1}{e}}$ at $x=1$; absolute minimum $0$ at the end point $x=0$.
• Example 13 Find the absolute extrema of $f(x)=\sin x+\cos x$ on $[0,2\pi ]$.
• Solution:
• $f^{\mathrm{\prime }}(x)=\cos x-\sin x=0$ gives $\tan x=1$, so $x={\displaystyle \frac{\pi }{4}}$ or ${\displaystyle \frac{5\pi }{4}}$.
• $f\left({\displaystyle \frac{\pi }{4}}\right)=\sqrt{2}$ and $f\left({\displaystyle \frac{5\pi }{4}}\right)=-\sqrt{2}$.
• End points f(0)=f(2π)=1.
• Absolute maximum $\sqrt{2}$ and absolute minimum $-\sqrt{2}$.
• Example 14 Find the absolute extrema of $f(x)=\sqrt{x-1}$ on $[2,5]$.
• Solution:
• $f^{\mathrm{\prime }}(x)={\displaystyle \frac{1}{2\sqrt{x-1}}}>0$, so there is no critical point and $f$ is increasing.
• The extrema are at the end points minimum f(2)=1 and maximum f(5)=2.

Remarks

• On a closed interval the end points must always be tested, because an extreme value can sit there with a non-zero derivative.
• A function may have several local extrema but only one absolute maximum value (it can be attained at more than one point).
• A local extremum is only "local"; the absolute extremum of an interval can coincide with a local extremum.