Find the maximum and minimum values attained by the function f along the path c(t). (a)f(x,y)=xy;c(t)=(cos⁡(t),sin⁡(t));0≤t≤2π maximum value__________ minimum value__________ (b) f(x,y)=x2+y2;c(t)=(cos⁡(t),8sin⁡(t));0≤t≤2π maximum value__________ minimum value__________

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Find the maximum and minimum values attained by the function f along the path c(t).   (a)f(x,y)=xy;c(t)=(cos⁡(t),sin⁡(t));0≤t≤2π   maximum value__________   minimum value__________  (b) f(x,y)=x2+y2;c(t)=(cos⁡(t),8sin⁡(t));0≤t≤2π  maximum value__________   minimum value__________

Paineow · Accepted Answer

Srep 1  (a)  Given that, f=(x,y)=xy;c(t)=(cos⁡t,sin⁡t);0≤t≤2π.  c(t)=xj^+yj^  =(cos⁡t)j^+(sin⁡t)j^  f(x,y)=cos⁡t×sin⁡  =sin⁡(2t)2  (∵sin⁡2α=2sin⁡αcos⁡α)  It is known that, the range of the sine functions is -1 to 1.  −1≤sin⁡(2t)≤1  −12≤sin⁡(2t)2≤12  −12≤f(x,y)≤12  Thus, the function has the maximum value 12 and minimum vaiue −12.

ramirezhereva · Accepted Answer

Step 2 (b) Given that, f(x,y)=x2+y2;c(t)=(cos⁡t,8sin⁡t);0≤t≤2π. c(t)=xj^+yj^ =(cos⁡t)j^+(8sin⁡t)j^ f(x,y)=(cos⁡t)2+(8sin⁡t)2 =cos2⁡t+64sin2=1−sin2⁡t+64sin2=1+63sin2 It is known that, the range of the sine function is -1 to 1. 0≤sin2t≤1 0≤63sin2t≤63 1≤1+63sin2t≤64 1≤f(x,y)≤64 Thus, the function has the maximum value 64 and minimum value 1.

Find the maximum and minimum values attained by the function f along t

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