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Find all local maximum and minimums of the graph. Approximate as needed.
The graph will have local maximum if it changes from increasing to decreasing and a local minimum if it changes from decreasing to increaing. The graph has a local maximum at \(\displaystyle (-1, 2)\). The graph has a local minimum at \(\displaystyle (-4.4,-1.00)\). The point \(\displaystyle (4, -7)\) is neither a local max or local min.
\begin{tikzpicture}[scale=.4, font=\Large]
\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-10,-10) grid (10,10);
\draw[latex-latex, very thick] (-10.4,0)--(10.4,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-10.4)--(0,10.4) node[above]{$y$};
\foreach \x in { -10,-8,...,10 }{
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\draw[thick, color=blue, domain={-10.0}:{-1.0}, color=red, very thick,latex-, samples =350] plot(\x, {7*(\x)^2/27 + 62*(\x)/27 + 109/27});
\draw[very thick, color=red,-latex] (-1, 2) -- (4, -7) --(10, -7);
\end{tikzpicture}
\begin{question}Find all local maximum and minimums of the graph. Approximate as needed.\newline\begin{tikzpicture}[scale=.4, font=\Large]
\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-10,-10) grid (10,10);
\draw[latex-latex, very thick] (-10.4,0)--(10.4,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-10.4)--(0,10.4) node[above]{$y$};
\foreach \x in { -10,-8,...,10 }{
\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
}
\foreach \y in { -10,-8,...,10 }{
\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\draw[thick, color=blue, domain={-10.0}:{-1.0}, color=red, very thick,latex-, samples =350] plot(\x, {7*(\x)^2/27 + 62*(\x)/27 + 109/27});
\draw[very thick, color=red,-latex] (-1, 2) -- (4, -7) --(10, -7);
\end{tikzpicture}
\soln{0cm}{The graph will have local maximum if it changes from increasing to decreasing and a local minimum if it changes from decreasing to increaing. The graph has a local maximum at $(-1, 2)$. The graph has a local minimum at $(-4.4,-1.00)$. The point $(4, -7)$ is neither a local max or local min. }
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
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<p> <p>The graph will have local maximum if it changes from increasing to decreasing and a local minimum if it changes from decreasing to increaing. The graph has a local maximum at <img class="equation_image" title=" \displaystyle (-1, 2) " src="/equation_images/%20%5Cdisplaystyle%20%28-1%2C%202%29%20" alt="LaTeX: \displaystyle (-1, 2) " data-equation-content=" \displaystyle (-1, 2) " /> . The graph has a local minimum at <img class="equation_image" title=" \displaystyle (-4.4,-1.00) " src="/equation_images/%20%5Cdisplaystyle%20%28-4.4%2C-1.00%29%20" alt="LaTeX: \displaystyle (-4.4,-1.00) " data-equation-content=" \displaystyle (-4.4,-1.00) " /> . The point <img class="equation_image" title=" \displaystyle (4, -7) " src="/equation_images/%20%5Cdisplaystyle%20%284%2C%20-7%29%20" alt="LaTeX: \displaystyle (4, -7) " data-equation-content=" \displaystyle (4, -7) " /> is neither a local max or local min. </p> </p>