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# challenging lagrange multipliers question?

Determine the regions of the pq-plane for which the function f(x,y) = 2px + qy^2 restricted to the curve x^3 + y^3 = 1 has exactly one and exactly three critical points, respectively

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- santmann2002Lv 79 years agoFavorite Answer
F(x,y,k)= 2px+qy^2+k(x^3+y^3-1)

Fx= 2p +3kx^2=0

Fy= 2qy+ +3ky^2=0

so y=0 and x=1 2q+3ky=0 so k= -2q/3y = -2p/3x^2

q/y=p/x^2 so y=qx^2/p and (q/p)^3*x^6+x^3-1=0

call x^3 = z

(p/q)^3z^2 +z-1=0 if 1+4(p/q)^3 >0 there are two roots for x^3 and so for x and added to (1,0) there are three critical points

If 1+4(p/q)^3=0 (p/q)^3= -1/4 and -1/4z^2+z-1 =z^2-4z+4= (z-2)^2=0 and z= 2 and x=2^1/3

if1+(p/q)^3<0 only one criticxal point (1,0)

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