Turning Points and Nature

A turning point of a function is a point where $f'(x)=0$. Turning Points A maximum turning point is a turning point where the curve is concave up (from increasing to decreasing ) and $f'(x)=0$ at the point.
$$ \begin{array}{|c|c|c|} \hline f'(x) \gt 0 & f'(x) = 0 & f'(x) \lt 0 \\ \hline & \text{maximum} & \\ \nearrow & & \searrow \\ \hline \end{array} $$
A minimum turning point is a turning point where the curve is concave down (from decreasing to increasing) and $f'(x)=0$ at the point.
$$ \begin{array}{|c|c|c|} \hline f'(x) \lt 0 & f'(x) = 0 & f'(x) \gt 0 \\ \hline \searrow & & \nearrow \\ & \text{minimum} & \\ \hline \end{array} $$

Example 1

Find any turning points and their nature of $f(x) =2x^3-9x^2+12x+3$.

Example 2

Find any turning points and their nature of $f(x) =-x^3+6x^2-9x-5$.

Determining the Nature of Turning Points by Concavities

\begin{array}{|c|c|} \hline \text{concave downwards} & \text{concave upwards} \\ \hline \cup \text{ shape} & \cap \text{ shape} \\ \hline f''(x) \gt 0 & f''(x) \lt 0 \\ \hline \end{array} A maximum turning point is a turning point where the curve is concave upwards, $f''(x) \lt 0$ and $f'(x)=0$ at the point.
A minimum turning point is a turning point where the curve is concave downwards, $f''(x) \gt 0$ and $f'(x)=0$ at the point.
$$ \begin{array}{|c|c|} \hline \text{maximum turning point} & \text{minimum turning point} \\ \hline f'(x) = 0 & f'(x) = 0 \\ \hline f''(x) \lt 0 & f''(x) \gt 0 \\ \hline \end{array} $$

Example 3

Find any turning points and their nature of $f(x) =2x^3-9x^2+12x+3$ using second derivatives.

Example 4

Find any turning points and their nature of $f(x) =-x^3+6x^2-9x-5$ using second derivatives.