\begin{itemize}
\item The $y$ values should be larger than $\ec m$, because $\ec m$ is a minimum.
\item If $\ec x<c$ but close to $\ec c$, the slopes of the secant lines will be negative:\[
		\ec\frac{f(x)-f(c)}{x-c}=\frac{\mathrm{positive}}{\mathrm{negative}}
	\]so the slopes of the tangent lines should
	also be negative. Thus the derivative should be negative for $\ec x<c$.
\end{itemize}



\pause{Extrema: Certainty of computation}
\noindent
Think about what must happen to the $y$ values around a minimum $\ec m=f(c)$ at $\ec x=a$.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.2]{figures/x_ex_right_decreasing.png}\end{center}
\end{center}\end{figure}
\begin{itemize}
\item The $y$ values should be larger than $\ec m$, because $\ec m$ is a minimum.
\item If $\ec x>c$ but close to $\ec c$, the slopes of the secant lines will be positive:\[
		\ec\frac{f(x)-f(c)}{x-c}=\frac{\mathrm{positive}}{\mathrm{positive}}
	\]so the slopes of the tangent lines should
	also be positive. Thus the derivative should be positive for $\ec x>c$.
\end{itemize}

