Local Maxima and Minima
A local maximum is a point that is higher than everything nearby — the top
of a hill. A local minimum is lower than everything nearby — the bottom of a
valley. Together they are called the local extrema of the function.
We already know that these turning points can only happen at
critical points. But not
every critical point is a peak or valley — so we need a test to tell which is which.
The first-derivative test
Walk left to right through a critical point and watch how the slope's sign changes:
- f' goes + → − (up then down): a local maximum \nearrow\searrow
- f' goes − → + (down then up): a local minimum \searrow\nearrow
- f' keeps the same sign: neither — the curve only paused
\overbrace{f' > 0}^{\nearrow}\;\big|\;\overbrace{f' < 0}^{\searrow} \;\Rightarrow\; \textbf{max} \qquad\qquad \overbrace{f' < 0}^{\searrow}\;\big|\;\overbrace{f' > 0}^{\nearrow} \;\Rightarrow\; \textbf{min}
See the sign flip
Here is f(x) = x^3 - 3x again. Slide the marker through
x = -1: the slope goes from positive to negative, so that's a
local max. Through x = 1 it goes from negative to
positive — a local min. The thin curve is f';
notice it crosses zero downward at the max and upward at the min.
A worked example
Take f(x) = x^3 - 3x. We found
f'(x) = 3(x-1)(x+1), critical at x = \pm 1.
Test a point in each interval:
f'(-2) = 9 > 0,\quad f'(0) = -3 < 0,\quad f'(2) = 9 > 0
Reading the signs +\,-\,+: at x=-1 the
slope flips +\to- (a local maximum, value
f(-1)=2); at x=1 it flips
-\to+ (a local minimum, value
f(1)=-2). Step the diagram below to build the verdict.
Watch it on Khan Academy