Turning Points and Their Nature

We can find the maximum, minimum, turning (or stationary) points of a function by differentiation. Given a graph  
, we can find  
  and solve  
. This will give us some values of  
. Substituting these values into the expression  
  will give us the  
  and allow us to write down the points  
To find if the point is a maximum or minimum we differentiate again to find  
, and substitute the relevant  
. 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  
  and determine the type of point.

We solve  
\[\frac{dy}{dx}=2x-8=0 \rightarrow x=4\]
\[y=x^2-8x+2=4^2-8 \times 4+2=-14\]
The turning or stationary point is  

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