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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
\(R\) is illustrated below. In this activity, you will carry out the steps to find the absolute maximum and absolute minimum of
\(f\) on
\(R\text{.}\)
(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
\(R\text{.}\) Write a couple of sentences to describe how the surface plot of
\(f\) shown below illustrates your results.
