anonymous
  • anonymous
find the maximum value of the function f(x,y,z)=x+2y+3z on the curve of interjection of the plane x-y+z=1 and cylinder x2+y2=1
Calculus1
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
find the maximum value of the function f(x,y,z)=x+2y+3z on the curve of interjection of the plane x-y+z=1 and cylinder x2+y2=1
amistre64
  • amistre64
i recall that the normal of a surface always points in the steepest direction
amistre64
  • amistre64
well, the gradF that is ...

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aroyni
  • aroyni
This is a constrained maximization problem. You are looking for the maximum of f constrained to the the two equation that you have given. I think the best way is to use Lagrange multipliers. Is this calculus 2? I can tell you more later if you want.
dumbcow
  • dumbcow
another approach is to recognize that x^2 +y^2 =1 is the unit circle the critical points of "f" are along outermost points of constrained region this represents a slanted circle where x,y are points along unit circle and z is defined by the given plane (z = 1+y-x) make substitution: \[x = \cos \theta\] \[y = \sin \theta\] \[z = 1+\sin \theta - \cos \theta\] now you can define "f" in terms of just angle "theta" and find angle which maximizes "f" \[f(\theta) = 5\sin \theta -2\cos \theta +3\] \[f'(\theta) = 5\cos \theta +2\sin \theta = 0\] \[\rightarrow \tan \theta = -\frac{5}{2}\] \[\theta_1 \approx 111.8\] \[\theta_2 \approx 291.8\] one will yield max the other min \[f(\theta_1) = 8.385\] \[f(\theta_2) = -2.385\] therefore maximum is 8.385 with angle of 111.8 degrees

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