91影视

The given function is $f(x,y)=\frac{x}{y}$and the boundary of the rectangle is$-1鈮�x鈮�1,1鈮�y鈮�4.$

Let's find the gradient of the given function,

$$\begin{gathered} \mathrm{鈭�}f(x,y)=\left(\frac{\mathrm{鈭�}}{\mathrm{鈭�}x}\frac{x}{y}\right)\mathrm{i}+\left(\frac{\mathrm{鈭�}}{\mathrm{鈭�}y}\frac{x}{y}\right)\mathrm{j} \\ \begin{array}{rl}\mathrm{鈭�}\mathrm{f}(\mathrm{x},\mathrm{y})& =\frac{1}{\mathrm{y}}\left(\frac{\mathrm{鈭�}}{\mathrm{鈭�}\mathrm{x}}\mathrm{x}\right)\mathrm{i}+\mathrm{x}\left(\frac{\mathrm{鈭�}}{\mathrm{鈭�}\mathrm{y}}\frac{1}{\mathrm{y}}\right)\mathrm{j}\\ \mathrm{鈭�}\mathrm{f}(\mathrm{x},\mathrm{y})& =\frac{1}{\mathrm{y}}\mathrm{i}+\mathrm{x}(鈭�1)\frac{1}{{\mathrm{y}}^2}\mathrm{j}\\ \mathrm{鈭�}\mathrm{f}(\mathrm{x},\mathrm{y})& =\frac{1}{\mathrm{y}}\mathrm{i}鈭�\frac{\mathrm{x}}{{\mathrm{y}}^2}\mathrm{j}\end{array} \end{gathered}$$

Now, to find the critical points put the above equation to zero.

So,

$$\begin{gathered} \begin{array}{rl}\mathrm{鈭�}\mathrm{f}(\mathrm{x},\mathrm{y})& =\frac{1}{\mathrm{y}}\mathrm{i}鈭�\frac{\mathrm{x}}{{\mathrm{y}}^2}\mathrm{j}\end{array} \\ \left(0i-0j\right)=\frac{1}{\mathrm{y}}\mathrm{i}鈭�\frac{\mathrm{x}}{{\mathrm{y}}^2}\mathrm{j} \\ 0=\frac{1}{\mathrm{y}}\mathrm{and}0=-\frac{\mathrm{x}}{{\mathrm{y}}^2} \end{gathered}$$

Thus, the point$\left(0,0\right)$is not considered where the maximum or minimum appears.

Now, let's take the boundary of the region when $x=1,1鈮�y鈮�4,$

$f(1,y)=g\left(y\right)=\frac{1}{y}$

Differentiate the above equation,

role="math" localid="1649952099684" $\begin{array}{r}\frac{d}{dy}g\left(y\right)=\frac{d}{dy}\frac{1}{y}\\ g\text{'}\left(y\right)=鈭�\frac{1}{y^2}\end{array}$

Since the function will become undefined when y = 0.

Now, at the endpoints of the region, the value of the function is,

$$\begin{gathered} g\left(1\right)=f(1,1)=1 \\ g\left(4\right)=f(1,4)=\frac{1}{4} \end{gathered}$$

Let's take the region when role="math" localid="1649952400385" $y=4,x=-1-1鈮�x鈮�1,$

$f(-1,4)=g(-1)=-\frac{1}{4}$

When role="math" localid="1649952560111" $x=-1,y=1$

role="math" localid="1649952580754" $f(-1,1)=g(-1)=-1$

Thus, the maximum value of the function is$1\mathrm{at}\left(1,1\right)$and the minimum value of the function is$-1\mathrm{at}\left(-1,1\right).$