Increasing and decreasing functions

Increasing functions. A function $f$ is said to be monotonically (or strictly) increasing on an interval $(a,b)$ if, for any $x_1,x_2\in(a,b)$ with $x_1<x_2$, we have $f(x_1)<f(x_2)$.

To connect this definition to the derivative: consider an increasing function $y=f(x)$ on $(a,b)$, and let $h>0$ be a small increment in $x$. Since $x<x+h$, and $f$ is increasing, $f(x)<f(x+h)$, so $f(x+h)-f(x)>0$, and dividing by the positive quantity $h$: $\dfrac{f(x+h)-f(x)}{h}>0$. Taking the limit as $h\to0$: $\displaystyle\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}\ge0$, i.e. $f'(x)\ge0$.

Moreover, if $f'(a)>0$, then in a small neighbourhood $(a-\delta,a+\delta)$ around $a$, $f$ is strictly increasing: for $0<h<\delta$, one can show $f(a-h)<f(a)<f(a+h)$.

Decreasing functions. A function $f$ is monotonically (strictly) decreasing on $(a,b)$ if, for any $x_1,x_2\in(a,b)$ with $x_1<x_2$, we have $f(x_1)>f(x_2)$. By an entirely analogous argument (now $f(x+h)<f(x)$ for $h>0$), this leads to $f'(x)\le0$ wherever $f$ is decreasing, and $f'(a)<0$ implies $f$ is strictly decreasing in a small neighbourhood of $a$.

Note: wherever $f'(x)=0$, the tangent is parallel to the X-axis, but this alone does not tell us whether $f$ is increasing or decreasing at that point — that must be decided from the behaviour of $f'$ on either side.

Worked Example 1. Show that $f(x)=x^3+10x+7$ is strictly increasing for all $x\in\mathbb{R}$. $f'(x)=3x^2+10$. Since $3x^2\ge0$ for all real $x$ and $10>0$, $f'(x)=3x^2+10>0$ everywhere, so $f$ is strictly increasing on all of $\mathbb{R}$.

Worked Example 2. Test whether $f(x)=x^3+6x^2+12x-5$ is increasing or decreasing for all $x\in\mathbb{R}$. $f'(x)=3x^2+12x+12=3(x+2)^2$. Since $3(x+2)^2\ge0$ for all $x$ (and is strictly positive except at the single point $x=-2$), $f'(x)\ge0$ everywhere, so $f$ is increasing for all $x\in\mathbb{R}$. …