If possible, find the absolute maximum and minimum values of the following function on the region .

Determine the absolute maximum value of on . Select the correct choice below and, if necessary, fill in the
answer box to complete your choice.
A. The absolute maximum value of on is
B. The function has no absolute maximum value on .

Ask by Lawson Simmons. in the United States

Mar 21,2025

Question

If possible, find the absolute maximum and minimum values of the following function on the region .

Determine the absolute maximum value of on . Select the correct choice below and, if necessary, fill in the
answer box to complete your choice.
A. The absolute maximum value of on is
B. The function has no absolute maximum value on .

Ask by Lawson Simmons. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

We are given $$ f(x,y)=4e^{-3x-3y} $$ with the region $$ R=\{(x,y): x\ge 0,\, y\ge 0\}. $$ **Step 1. Analyze the function.** Since the exponential function is strictly decreasing as its exponent increases, and $$ -3x-3y $$ is maximized (i.e., least negative) when $$ x $$ and $$ y $$ are as small as possible, the maximum value is achieved when both $$ x=0 $$ and $$ y=0 $$. **Step 2. Compute the value at $$ (0,0) $$:** Substitute into $$ f(x,y) $$: $$ f(0,0)=4e^{-3(0)-3(0)}=4e^0=4. $$ **Step 3. Comparison over the region $$ R $$.** For any point $$(x,y)\in R$$ where either $$ x>0 $$ or $$ y>0 $$, the expression $$-3x-3y$$ is less than zero. Consequently, $$ e^{-3x-3y}<1 $$, and thus $$ f(x,y)=4e^{-3x-3y}<4. $$ So the maximum value of $$ f $$ on $$ R $$ is $$ 4 $$. **Conclusion:** The absolute maximum value of $$ f(x,y) $$ on the region $$ R $$ is $$ 4 $$, and it occurs at $$ (0,0) $$. Select the correct choice: A. The absolute maximum value of $$ f $$ on $$ R $$ is $$ 4 $$. The absolute maximum value of $$ f(x, y) $$ on $$ R $$ is 4.

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