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mars1129 [50]
3 years ago
11

Find the maximum and minimum values attained by f(x, y, z) = 5xyz on the unit ball x2 + y2 + z2 ≤ 1.

Mathematics
1 answer:
Allushta [10]3 years ago
5 0
Check for critical points within the unit ball by solving for when the first-order partial derivatives vanish:
f_x=5yz=0\implies y=0\text{ or }z=0
f_y=5xz=0\implies x=0\text{ or }z=0
f_z=5xy=0\implies x=0\text{ or }y=0


Taken together, we find that (0, 0, 0) appears to be the only critical point on f within the ball. At this point, we have f(0,0,0)=0.

Now let's use the method of Lagrange multipliers to look for critical points on the boundary. We have the Lagrangian

L(x,y,z,\lambda)=5xyz+\lambda(x^2+y^2+z^2-1)

with partial derivatives (set to 0)

L_x=5yz+2\lambda x=0
L_y=5xz+2\lambda y=0
L_z=5xy+2\lambda z=0
L_\lambda=x^2+y^2+z^2-1=0

We then observe that

xL_x+yL_y+zL_z=0\implies15xyz+2\lambda=0\implies\lambda=-\dfrac{15xyz}2

So, ignoring the critical point we've already found at (0, 0, 0),


5yz+2\left(-\dfrac{15xyz}2\right)x=0\implies5yz(1-3x^2)=0\implies x=\pm\dfrac1{\sqrt3}
5xz+2\left(-\dfrac{15xyz}2\right)y=0\implies5xz(1-3y^2)=0\implies y=\pm\dfrac1{\sqrt3}
5xy+2\left(-\dfrac{15xyz}2\right)z=0\implies5xy(1-3z^2)=0\implies z=\pm\dfrac1{\sqrt3}

So ultimately, we have 9 critical points - 1 at the origin (0, 0, 0), and 8 at the various combinations of \left(\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3}\right), at which points we get a value of either of \pm\dfrac5{\sqrt3}, with the maximum being the positive value and the minimum being the negative one.
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