## Stationary points

To find the stationary or turning points of a curve
$y=f(x)$
, solve
$\frac{d(f(x))}{dx}=0$
.
To classify them, find
$\frac{d^2 (f(x))}{dx^2}$
. If
$\frac{d^2 (f(x))}{dx^2} \gt 0$
at the point, the point is a minimum and if
$\frac{d^2 (f(x))}{dx^2} \lt 0$
at the point, the point is a maximum. If
$\frac{d^2 (f(x))}{dx^2}=0$
, the the point is a stationary point of inflexion.
Example:
$f(x)=3x^5-5x^3$
.
$\frac{d(f(x))}{dx}15x^4-15x^2=0 \rightarrow 15x^2(x^2-1)=0 \rightarrow x=-1, \; 0, \; 1$
.
$\frac{d^2 (f(x))}{dx^2}=60x^3-30x$

At
$x=-1$
,
$\frac{d^2(f(x))}{dx^2}=-30 \lt 0$
so this point is a maximum.
At
$x=0$
,
$\frac{d^2(f(x))}{dx^2}= 0$
so this point is a stationary point of inflexion.
At
$x=1$
,
$\frac{d^2(f(x))}{dx^2}=30 \gt 0$
so this point is a minimum.