Guest boysetsfire Posted April 1, 2004 Share Posted April 1, 2004 I had this question on a test today and just want to know if i was on the right path untill i got too frustrated and gave up. Find the closest points to the origin on the surface x^2-yz=5. This is what i did d=sqrt( (x-0)^2 + ( y-0 )^2 + (z-0 )^2 ) then i took the f(x,y) = x^2 + y^2 +( (x^2-5)/y )^2 took partial derivatives but then i couldnt solve for either x or y. -Marc Link to comment Share on other sites More sharing options...
wolfson Posted April 1, 2004 Share Posted April 1, 2004 D^2 = x^2 + y^2 + z^2 = y^2 + y^z + z^2 + 5 Now note that if z is non-zero then the expression (y^2 + y^z + z^2) is always greater than zero, Therefore the smallest value of D^2 (and hence D) will be when z = y = 0. Thus finding the partial derivitives of this expression w.r.t. to y and z and setting them equal to 0. Threrfore 5,0,0. Oh and note also that (-5,0,0) is also a point. Link to comment Share on other sites More sharing options...
wolfson Posted April 2, 2004 Share Posted April 2, 2004 Oops... mistake The points should be (sqroot5,0,0) and (-sqroot5,0,0). Link to comment Share on other sites More sharing options...
bloodhound Posted April 13, 2004 Share Posted April 13, 2004 for general questions like this, u can use the method of lagrange multipliers. basically if u are given a function f and u have to minimise or maximise the value of f given a condition that another function g=0 then if u form another function F(x,y,z,lamda)=f(x,y,z)-lamda*g(x,y,z) and then u find the partial derivatives , Fx,Fy,Fz,Flamnda, and solve Fx=Fy=Fz=Flamnda=0 , then those values of (x,y,z) will minimise the value of f(x,y,z) with the condition that g(x,y,z)=0 In this case we are minimising (x^2+y^2+z^2)^(1/2) which is the same as minimising(x^2+y^2+z^2) set that = f(x,y,z) . also we are given the condition that g(x,y,z)=x^2-yz-5=0 now take F(x,y,z,lamda)=f(x,y,z)-lamda*g(x,y,z) =x^2+y^2+z^2 - lamda*(x^2-yz-5) so now find all the partial derivatives. equate them to 0 and solve them. that is ur values which will minimise ur function This method can be applied to function of as many variables as u like with as many conditions eg. minimise f(r,s,t,v) given g(r,s,t,v)=0 and h(r,s,t,v)=0 u create F(r,s,t,v,lambda, mu)=f(r,s,t,v)-lambda*g(r,s,t,v) - mu*h(r,s,t,v) u find the partial derivatives of F w.r.t r,s,t,v, lambda and mu and equate them to 0 and solve them. Those valuesof (r,s,t,v) minimises or maximises the function f(r,s,t,v) and so it can be extended to functions of as many variables as u like 1 Link to comment Share on other sites More sharing options...
bloodhound Posted April 13, 2004 Share Posted April 13, 2004 ill just do the given question for an example we have F(x,y,z,lamda)=x^2+y^2+z^2 - lamda*(x^2-yz-5) finding the four partial derivatives and equating them to 0 we get 4 simultaneous equations. 2x-2lambda*x=0 which gives us lambda = 1 partial dev wrt y gives us 2y+lambda*z=0 since lambda = 1 we get 2y+z=0 partial dev wrt z gives us 2z+lambda*y=0 since lambda = 1 we get 2z+y=0 therefore from the last two equations , the only solution is y=0,z=0 Now, Partial Dev wrt lambda gives us 5-x^2=0 which gives us x=plus.minus(sqrt(5)) Therefore (x,y,z)=(+-sqrt(5),0,0) minimises the distance from the origin given =0 But if we look at x^2+y^2+z^2 we see that its nonnegative and not bounded above there fore the points(+-sqrt(5),0,0) minimises the distance Link to comment Share on other sites More sharing options...
wolfson Posted April 13, 2004 Share Posted April 13, 2004 Thats they way Link to comment Share on other sites More sharing options...
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