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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Activity 11.9.5 .
Let
\(f(x,y) = x^2-3y^2-4x+6y\) with triangular domain
\(R\) whose vertices are at
\((0,0)\text{,}\) \((4,0)\text{,}\) and
\((0,4)\text{.}\) The domain
\(R\) appears in
Figure 11.9.17 . In this activity, we will go through the steps to find the absolute maximum and minimum of
\(f\) on
\(R\text{.}\)
Figure 11.9.17. A plot of \(R\text{,}\) the domain of \(f(x,y) = x^2-3y^2-4x+6y\)
(a) Find all of the critical points of
\(f\) in
\(R\text{.}\)
(b) Parameterize the edge of
\(R\) that is on the
\(x\) -axis and find the critical points of
\(f\) on that edge.
Hint .
You may need to consider endpoints
(c) Parameterize the edge of
\(R\) that is on the
\(y\) -axis and find the critical points of
\(f\) on that edge.
Hint .
You may need to consider endpoints
(d) Parameterize the diagonal edge of
\(R\) and find the critical points of
\(f\) on that edge.
Hint .
You may need to consider endpoints
(e) Find the absolute maximum and absolute minimum values of
\(f\) on
\(R\) and write a couple of sentences to compare your results to the surface plot
Figure 11.9.18.
