How the Sign of a Derivative Tells You Whether a Graph Is Climbing

A share price chart, a fever chart and a reservoir level record all rise in some stretches and fall in others. The first derivative detects this exactly: its sign tells whether a function is going up or down, without plotting a single point.

This lesson covers the first derivative and intervals of increase and decrease, proving that a function increases or decreases, and applied problems.

**A function is increasing on an interval where $f^{\prime }(x)>0$ and decreasing where $f^{\prime }(x)<0$, so its intervals of increase and decrease are found by locating the points where $f^{\prime }(x)=0$ or is undefined and testing the sign of $f^{\prime }(x)$ between them.

Steps:**

- Find $f^{\prime }(x)$ and solve $f^{\prime }(x)=0$
- Mark these points on a number line to form intervals
- Test the sign of $f^{\prime }(x)$ in each interval

Worked example. Where is $f(x)=2x^3-9x^2+12x+5$ increasing or decreasing?

- $f^{\prime }(x)=6x^2-18x+12=6(x-1)(x-2)$
- For $x<1$: $f^{\prime }(0)=12>0$, so increasing
- For $1<x<2$: $f^{\prime }(1.5)=-1.5<0$, so decreasing
- For $x>2$: $f^{\prime }(3)=12>0$, so increasing

So f increases on $(-\infty ,1]$ and $[2,\infty )$ and decreases on $[1,2]$.

An everyday example. A child's temperature chart during a fever climbs while the fever builds and falls once medicine works — the sign of its rate of change shows which phase it is in.

The substance. A zero derivative at a single point does not stop a function from increasing — $x^3$ has zero slope at 0 but increases on all of $\mathbb{R}$.

**To prove that f is increasing on an interval, show that $f^{\prime }(x)>0$ — or $f^{\prime }(x)\ge 0$ with equality only at isolated points — for every x in the interval, often by writing $f^{\prime }(x)$ as a perfect square or by using bounds such as $-1\le \cos x\le 1$.

Worked example.** Prove that $f(x)=x^3-3x^2+3x+7$ is increasing on $\mathbb{R}$.

$$f^{\prime }(x)=3x^2-6x+3=3(x-1{)}^2\ge 0$$

with equality only at $x=1$, so f is increasing everywhere.

Worked example 2. For $f(x)=\log (\sin x)$, $f^{\prime }(x)=\cot x$, which is positive on $\left(0,\frac{\pi }{2}\right)$ and negative on $\left(\frac{\pi }{2},\pi \right)$, so f increases on the first interval and decreases on the second.

Worked example 3. For $f(x)=x-\sin x$, $f^{\prime }(x)=1-\cos x\ge 0$, with equality only at the isolated points $x=2n\pi$, so f is increasing on $\mathbb{R}$.

Proving inequalities. Since $x-\sin x$ is increasing and equals 0 at $x=0$, it is positive for $x>0$, so $\sin x<x$ for every positive x.

An everyday example. A savings account that only ever receives deposits has a balance that never goes down, even on days with no transaction.

The substance. The sign of the derivative must hold across the whole interval — checking a few points proves nothing.

Applied problems are solved by writing the quantity as a function of one variable, finding where its derivative is positive or negative, and interpreting those intervals in context — such as when profit rises or when a particle moves backwards.

Worked example (business). A sweet stall's daily profit from selling x kg is $P(x)=240x-2x^2-1000$ rupees. Then $P^{\prime }(x)=240-4x$, which is positive for $x<60$ and negative for $x>60$. Profit rises until 60 kg are sold and falls after that: $P(60)=6200$, while $P(50)$ and $P(70)$ are both 6000.

Worked example (motion). A particle has position $s=t^3-9t^2+24t$ metres. Its velocity $v=3t^2-18t+24=3(t-2)(t-4)$ is positive for $t<2$ and $t>4$, when it moves forward, and negative for $2<t<4$, when it moves backwards.

Worked example (geometry). A rectangle with perimeter 20 cm and one side x has area $A=x(10-x)$; since $A^{\prime }=10-2x$, the area increases while $x<5$ and decreases after.

An everyday example. A farmer tracking onion prices through a season wants the intervals when prices rise, to decide the best time to sell.

The substance. The point where a function stops increasing and starts decreasing is a local maximum — the idea developed fully in the next chapter.