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soldi70 [24.7K]
3 years ago
10

The equations that must be solved for maximum or minimum values of a differentiable function w=​f(x,y,z) subject to two constrai

nts ​g(x,y,z)=0 and ​h(x,y,z)=​0, where g and h are also​ differentiable, are gradientf=lambdagradientg+mugradient​h, ​g(x,y,z)=​0, and ​h(x,y,z)=​0, where lambda and mu ​(the Lagrange​ multipliers) are real numbers. Use this result to find the maximum and minimum values of ​f(x,y,z)=xsquared+ysquared+zsquared on the intersection between the cone zsquared=4xsquared+4ysquared and the plane 2x+4z=2.
Mathematics
1 answer:
sammy [17]3 years ago
5 0

The Lagrangian is

L(x,y,z,\lambda,\mu)=x^2+y^2+z^2+\lambda(4x^2+4y^2-z^2)+\mu(2x+4z-2)

with partial derivatives (set equal to 0)

L_x=2x+8\lambda x+2\mu=0\implies x(1+4\lambda)+\mu=0

L_y=2y+8\lambda y=0\implies y(1+4\lambda)=0

L_z=2z-2\lambda z+4\mu=0\implies z(1-\lambda)+2\mu=0

L_\lambda=4x^2+4y^2-z^2=0

L_\mu=2x+4z-2=0\implies x+2z=1

Case 1: If y=0, then

4x^2-z^2=0\implies4x^2=z^2\implies2|x|=|z|

Then

x+2z=1\implies x=1-2z\implies2|1-2z|=|z|\implies z=\dfrac25\text{ or }z=\dfrac23

\implies x=\dfrac15\text{ or }x=-\dfrac13

So we have two critical points, \left(\dfrac15,0,\dfrac25\right) and \left(-\dfrac13,0,\dfrac23\right)

Case 2: If \lambda=-\dfrac14, then in the first equation we get

x(1+4\lambda)+\mu=\mu=0

and from the third equation,

z(1-\lambda)+2\mu=\dfrac54z=0\implies z=0

Then

x+2z=1\implies x=1

4x^2+4y^2-z^2=0\implies1+y^2=0

but there are no real solutions for y, so this case yields no additional critical points.

So at the two critical points we've found, we get extreme values of

f\left(\dfrac15,0,\dfrac25\right)=\dfrac15 (min)

and

f\left(-\dfrac13,0,\dfrac23\right)=\dfrac59 (max)

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