Turning Points and Nature


A turning point of a function is a point where $f'(x)=0$.

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.





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