Increasing and Decreasing Functions

MATIncreasing & Decreasing Functions

The sign of the first derivative determines whether a function rises or falls on an interval: $f'(x)>0$ means strictly increasing, $f'(x)<0$ means strictly decreasing. The standard task is to find $f'(x)$ , locate the critical points where $f'(x)=0$ or is undefined, mark them on a number line, and test the sign of $f'$ in each resulting interval.

This monotonicity analysis is closely tied to maxima/minima and is a quick, high-yield exam item.

Increasing function

f'(x)>0\ \forall x \in I \Rightarrow f \text{ strictly increasing on } I

I

is an interval; allowing

f'(x)=0

at isolated points still gives (non-strictly) increasing.

Decreasing function

f'(x)<0\ \forall x \in I \Rightarrow f \text{ strictly decreasing on } I

I

is an interval; the inequality is reversed for decreasing behaviour.

Critical points partition the line

f'(x)=0 \text{ or } f'(x)\text{ undefined}

These

x

-values split the domain into test intervals on which

f'

keeps a constant sign.

Method: compute $f'(x)$ , solve $f'(x)=0$ , place roots (and points of non-differentiability) on a number line, then test the sign of $f'$ in each interval.
Strictly increasing/decreasing uses strict inequalities $>0$ or $<0$ ; a function may be non-strictly monotonic if $f'=0$ only at isolated points.
To prove a function is increasing for ALL real $x$ , show $f'(x)\ge 0$ everywhere, often by writing $f'(x)$ as a perfect square, e.g. $f'(x)=3(x-1)^2\ge0$ .
Always state the answer over the domain of $f$ ; exclude points where $f$ is undefined, such as $x=0$ for $f(x)=x+\dfrac{1}{x}$ .
Factor $f'(x)$ fully so each interval's sign comes from the product of factor signs.
For $f(x)=\log(1+x)-\dfrac{x}{1+x}$ , $f'(x)=\dfrac{x}{(1+x)^2}\ge0$ on $[0,\infty)$ , proving it increases there.
Endpoints can be included for closed intervals when the one-sided behaviour matches.

Where the marks go

Including points outside the domain in an interval of increase/decrease (e.g. straddling $x=0$ for $x+\dfrac{1}{x}$ ).
Mixing up the direction: $f'(x)>0$ is increasing, not decreasing.
Reporting only the roots of $f'(x)=0$ instead of the intervals between them.
Failing to test the sign of $f'$ in each interval and instead guessing the monotonic behaviour.

How the board asks it

Numericalsign of $f'(x)$ across number-line intervals
Find the intervals in which the function $f(x)=2x^3-9x^2+12x+15$ is (i) strictly increasing and (ii) strictly decreasing.
Derive / proveshowing $f'(x)\ge 0$ as a perfect square over the domain
Prove that the function $f(x)=x^3-3x^2+3x-100$ is increasing for all real values of $x$ .
Numericalstating the answer over the domain of $f$ , excluding undefined points
Find the intervals in which $f(x)=x+\dfrac{1}{x}$ , $x\ne 0$ , is increasing and decreasing, stating clearly the domain over which your answer holds.
Numericalsign of $f'(x)$ on a restricted interval such as $[0,2\pi]$
Determine the intervals in which the function $f(x)=\sin x+\cos x$ , $0\le x\le 2\pi$ , is strictly increasing or strictly decreasing.
Numericalimposing $f'(x)\ge 0$ for all $x$ via the discriminant condition
Find the values of $k$ for which the function $f(x)=kx^3-9x^2+9x+3$ is increasing for all real values of $x$ .
Numericalusing $f'(x)\ge 0$ to establish an inequality
Show that the function $f(x)=\log(1+x)-\dfrac{x}{1+x}$ is increasing for all $x>-1$ , and hence prove that $\log(1+x)>\dfrac{x}{1+x}$ for $x>0$ .

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.