Module 4 · Lesson 4.5

Extrema & the Shape of a Curve

Find peaks and valleys by locating where the slope hits zero, then read the sign of the derivative on either side to classify each critical point.

Intuition

Walk along a hilly path. At a peak the ground levels out for an instant before sloping down; at a valley it levels out before sloping up. Those level spots are the places where the derivative is zero. They are the candidates for local maximum and minimum values.

A stock-price chart shows the same geometry: a local high is a moment when the instantaneous rate of change crosses from positive to negative. Traders watch those turning points; calculus locates them systematically.

Core concept

Critical points occur where $f'(x)=0$ or $f'$ is undefined. The first-derivative test examines the sign of $f'$ immediately to the left and right of each critical point: positive-to-negative means a local max, negative-to-positive a local min, same sign on both sides means neither.

$f(x)=x^{3}-3x$$f'=3x^{2}-3$
Critical pts$x=\pm 1$
At $x=-1$$f'=+$ to $-$ → local max
At $x=1$$f'=-$ to $+$ → local min
Values$f(-1)=2$, $f(1)=-2$

Interactive Lab

Move the probe along the curve. Watch the tangent slope change sign at the critical points and read the local-max / local-min classification.

$f(x)$: 0.00 $f'(x)$: −3.00
x y
Critical points of $f(x)=x^{3}-3x$ are at $x=\pm 1$.

Green marker is the local maximum; amber is the local minimum. The tangent is horizontal exactly at those points, and the sign change of $f'$ confirms the type of extremum.

Key Idea

$$ f'(c)=0 \quad\text{and}\quad f'\ \text{changes sign at }c $$

A number $c$ in the domain is a critical point when $f'(c)=0$ or $f'(c)$ does not exist. If $f'$ changes from positive to negative at $c$, then $f(c)$ is a local maximum; if it changes from negative to positive, $f(c)$ is a local minimum. No sign change means the critical point is neither a peak nor a valley (a horizontal inflection or a flat shoulder).

$c$A critical point where $f'(c)=0$ or is undefined
$f'(c)=0$The tangent is horizontal at that $x$-value
sign changeThe first-derivative test that classifies max vs min
local max/minHighest or lowest value in some open neighborhood of $c$

Think Further

  1. Can a function have a horizontal tangent yet neither a local max nor a local min? Give a simple example.
  2. On a stock chart, why might a local maximum be more useful to a short-term trader than the global maximum over a whole year?
  3. If $f'$ is zero on an entire interval, what does the graph look like there, and are those points still called critical points?