## Turning Points and Their Nature

We can find the maximum, minimum, turning (or stationary) points of a function by differentiation. Given a graph
$y=f(x)$
, we can find
$\frac{dy}{dx}$
and solve
$\frac{dy}{dx}=0$
. This will give us some values of
$x$
. Substituting these values into the expression
$y=f(x)$
will give us the
$y$
and allow us to write down the points
$(x,y)$
.
To find if the point is a maximum or minimum we differentiate again to find
$\frac{d^2y}{dx^2}$
, and substitute the relevant
$x$
. If the result is positive, the point is a minimum. If the result is negative, the result is a maximum.

Example: Find the turning point(s) of
$y+x^2-8x+2$
and determine the type of point.
$\frac{dy}{dx}=2x-8$

We solve
$\frac{dy}{dx}=2x-8=0 \rightarrow x=4$
.
Then
$y=x^2-8x+2=4^2-8 \times 4+2=-14$
.
The turning or stationary point is
$(4,-14)$

$\frac{d^2y}{dx^2}=2 \gt 0$
, so the point is a minimum.