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Ex 3:  If $f'(x) = 0$ for all $x \in (a, b)$, then $f(x) = c$ for some constant $c$.
\vfill
\vfill
Ex 4:  If $f'(x) = g'(x)$ for all $x \in (a, b)$, then $f(x) = g(x) + c$ for some constant $c$.
\vfill
\eject


{\it Increasing/Decreasing Test}: \hfil \break
If $f'(x) > 0$ for all $x \in (a, b)$, then $f$ is increasing on $(a, b)$

If $f'(x) < 0$ for all $x \in (a, b)$, then $f$ is decreasing on $(a, b)$

{\it First derivative test:}  \hfil \break
Suppose $c$ is a critical number of a continuous function $f$, then

\vfil

Defn:  $f$ is {\bf concave down} if the graph of $f$ \hfil \break
lies below the tangent lines to $f$.

Defn:  $f$ is {\bf concave up} if the graph of $f$ \hfil \break
lies above the tangent lines to $f$.

{\it Concavity Test}: \hfil \break
If  $f''(x) > 0$ for all $x \in (a, b)$, then $f$ is concave upward on $(a, b)$.

If $f''(x) < 0$ for all $x \in (a, b)$,  then $f$ is concave down on $(a, b)$.


Defn:  The point $(x_0, y_0)$ is an {\bf inflection point} if $f$ is continuous at $x_0$ and if the concavity changes at $x_0$
\vfil

{\it Second derivative test:} If $f''$ continuous at $c$, then \hfil \break  
If $f'(c) = 0$ and $f''(c) > 0$, then $f$ has a local minimum at $c$.
\vfil

If $f'(c) = 0$ and $f''(c) < 0$, then $f$ has a local maximum at $c$.

\vfil
If $f'(c) = 0$ and $f''(c) = 0$, second derivative test gives no info.

\eject



Converses are not true:

Increasing/Decreasing  Test

If $f'(x) > 0$ for all $x \in (a, b)$, then $f$ is increasing on $(a, b)$

$f$ increasing on $(a, b)$ does not imply $f'(x) >0$ for all $x \in (a, b)$.

Ex:

\vskip 30pt

If $f'(x) < 0$ for all $x \in (a, b)$, then $f$ is decreasing on $(a, b)$

$f$ decreasing on $(a, b)$ does not imply $f'(x) < 0$ for all $x \in (a, b)$.

Ex:

\vskip 30pt

{\it Concavity Test}: \hfil \break
If  $f''(x) > 0$ for all $x \in (a, b)$, then $f$ is concave upward on $(a, b)$.

$f$ concave upward on $(a, b)$ does not imply $f''(x) > 0$ for all $x \in (a, b)$.

Ex:

\vskip 30pt


If $f''(x) < 0$ for all $x \in (a, b)$,  then $f$ is concave down on $(a, b)$.

$f$ concave downward on $(a, b)$ does not imply $f''(x) < 0$ for all $x \in (a, b)$.

Ex:

\vskip 30pt


\end


\end