## 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.