Lecture 10 · Barnett ch. 4 · about 45 min

First Derivative and Graphs

Reading a function's shape from the sign of its derivative: where it rises and falls, its critical points, and the first-derivative test that tells local maxima from local minima.

By the end you can

Determine intervals where a function is increasing or decreasing from the sign of f prime
Find the critical numbers of a function
Apply the first-derivative test to classify each critical point
Sketch the shape of a profit or cost function using a sign chart

The sign of the derivative tells you the shape

The derivative is a rate of change. Its sign at a point tells you whether the function is rising or falling there, which is enough information to sketch the overall shape of a curve without plotting dozens of points.

Increasing and decreasingRulesFormula sheet →

f'(x) \gt 0 \text{ on an interval} \;\Rightarrow\; f \text{ is increasing there} \qquad f'(x) \lt 0 \;\Rightarrow\; f \text{ is decreasing there}

$f'(x)$: the derivative of f, whose sign on an interval tells you the direction f is moving

Use it when you need to describe where a function rises or falls, such as where profit is growing versus shrinking.

Critical numbers

A critical number is a value in the domain of $f$ where $f'(x) = 0$ or $f'(x)$ does not exist. These are the only candidates for local maxima and minima.

Critical numbersDefinitionFormula sheet →

f'(c) = 0 \quad \text{or} \quad f'(c) \text{ does not exist, with } c \text{ in the domain of } f

$c$: a critical number of f

Use it when you are looking for possible local maxima or minima of a function, as the first step before the first-derivative test.

The first-derivative test

Once you have the critical numbers, check the sign of $f'$ on each side of each one. A sign change tells you what kind of turning point it is.

First-derivative testKey ruleFormula sheet →

f' : + \to - \text{ at } c \;\Rightarrow\; \text{local max at } c \qquad f' : - \to + \text{ at } c \;\Rightarrow\; \text{local min at } c

$+ \to -$: f' is positive just before c and negative just after
$- \to +$: the reverse

Use it when you have found critical numbers and need to classify each one, or a question directly asks for local extrema.

If $f'$ does not change sign at $c$ (say, positive on both sides), then $c$ is neither a local max nor a local min — often a point where the curve flattens momentarily and keeps going the same direction, such as $f(x) = x^3$ at $x = 0$ .

Increasing/decreasing intervals in one picture

Exam practice

Summary and review

$f'(x) \gt 0$ : increasing. $f'(x) \lt 0$ : decreasing.
A critical number is where $f'(c) = 0$ or $f'(c)$ does not exist.
First-derivative test: sign change $+ \to -$ is a local max; $- \to +$ is a local min; no change means neither.
A critical number is only a candidate; the sign test (or the next lesson’s second-derivative test) is required to classify it.
Building a sign chart across all critical numbers gives the full increasing/decreasing shape of the function.