Find the exact extreme values of the function =f(x, y) = (x- subject to the foll
ID: 2885238 • Letter: F
Question
Find the exact extreme values of the function =f(x, y) = (x- subject to the following constraints 0S 16 0s ys 14 Start by listing all nine candidates, including their z values, in the form (x,y,z) First, list the four corner points and order your answers from smallest to largest x, then from smallest to largest y Next find the critical point. Lastly, find the four boundary points and order your answers from smallest to largest x, then from smallest to largest y Finally, find the extreme values minat (x,y),) f,nar= at (x,y)=(Explanation / Answer
1)(0,0,93)
2)(0,14,261)
3)(16,0,125)
4)(16,14,293)
5)(7,1,43)
on boundary ,x=0
f(y)=(0-7)2+(y-1)2+43
=>f '(y)=0+2(y-1)+0
=>f '(y)=2(y-1)
for extrema f '(y)=0
=>2(y-1)=0
=>y=1
a point on boundary is (x,y)=(0,1)
f(0,1)=92
on boundary ,x=16
f(y)=(16-7)2+(y-1)2+43
=>f '(y)=0+2(y-1)+0
=>f '(y)=2(y-1)
for extrema f '(y)=0
=>2(y-1)=0
=>y=1
a point on boundary is (x,y)=(16,1)
f(16,1)=124
on boundary ,y=0
f(x)=(x-7)2+(0-1)2+43
=>f '(x)=2(x-7)+0+0
=>f '(x)=2(x-7)
for extrema f '(x)=0
=>2(x-7)=0
=>x=7
a point on boundary is (x,y)=(7,0)
f(7,0)=44
on boundary ,y=14
f(x)=(x-7)2+(14-1)2+43
=>f '(x)=2(x-7)+0+0
=>f '(x)=2(x-7)
for extrema f '(x)=0
=>2(x-7)=0
=>x=7
a point on boundary is (x,y)=(7,14)
f(7,14)=212
6)(0,1,92)
7)(7,0,44)
8)(7,14,212)
9)(16,1,124)
fmin=43 at (x,y)=(7,1)
fmax=293 at (x,y)=(16,14)
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