 # Find the maximum and minimum values attained by the function f along t aspifsGak5u 2021-12-14 Answered
Find the maximum and minimum values attained by the function f along the path $c\left(t\right)$.
$\left(a\right)f\left(x,y\right)=xy;c\left(t\right)=\left(\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right)\right);0\le t\le 2\pi$
maximum value__________
minimum value__________
(b) $f\left(x,y\right)={x}^{2}+{y}^{2};c\left(t\right)=\left(\mathrm{cos}\left(t\right),8\mathrm{sin}\left(t\right)\right);0\le t\le 2\pi$
maximum value__________
minimum value__________
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Srep 1
(a)
Given that, $f=\left(x,y\right)=xy;c\left(t\right)=\left(\mathrm{cos}t,\mathrm{sin}t\right);0\le t\le 2\pi$.
$c\left(t\right)=x\stackrel{^}{j}+y\stackrel{^}{j}$
$=\left(\mathrm{cos}t\right)\stackrel{^}{j}+\left(\mathrm{sin}t\right)\stackrel{^}{j}$
$f\left(x,y\right)=\mathrm{cos}t×\mathrm{sin}$
$=\frac{\mathrm{sin}\left(2t\right)}{2}$
$\left(\because \mathrm{sin}2\alpha =2\mathrm{sin}\alpha \mathrm{cos}\alpha \right)$
It is known that, the range of the sine functions is -1 to 1.
$-1\le \mathrm{sin}\left(2t\right)\le 1$
$\frac{-1}{2}\le \frac{\mathrm{sin}\left(2t\right)}{2}\le \frac{1}{2}$
$\frac{-1}{2}\le f\left(x,y\right)\le \frac{1}{2}$
Thus, the function has the maximum value $\frac{1}{2}$ and minimum vaiue $-\frac{1}{2}$.

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Step 2
(b)
Given that, $f\left(x,y\right)={x}^{2}+{y}^{2};c\left(t\right)=\left(\mathrm{cos}t,8\mathrm{sin}t\right);0\le t\le 2\pi$.
$c\left(t\right)=x\stackrel{^}{j}+y\stackrel{^}{j}$
$=\left(\mathrm{cos}t\right)\stackrel{^}{j}+\left(8\mathrm{sin}t\right)\stackrel{^}{j}$
$f\left(x,y\right)={\left(\mathrm{cos}t\right)}^{2}+{\left(8\mathrm{sin}t\right)}^{2}$
$={\mathrm{cos}}^{2}t+64{\mathrm{sin}}^{2}\phantom{\rule{0ex}{0ex}}=1-{\mathrm{sin}}^{2}t+64{\mathrm{sin}}^{2}\phantom{\rule{0ex}{0ex}}=1+63{\mathrm{sin}}^{2}$
It is known that, the range of the sine function is -1 to 1.
$0\le {\mathrm{sin}}^{2}t\le 1$
$0\le 63{\mathrm{sin}}^{2}t\le 63$
$1\le 1+63{\mathrm{sin}}^{2}t\le 64$
$1\le f\left(x,y\right)\le 64$
Thus, the function has the maximum value 64 and minimum value 1.

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