Question

find the maximum value of x^m y^n z^p given that x+y+z=a

Answer 1

Given : $x^m.y^n.z^p$

x+y+z=a

To Find : maximum value

Solution:

Let say  U =  $x^m.y^n.z^p$

∂U/∂x     $= mx^{m-1}.y^n.z^p$

∂U/∂y     $= nx^{m}.y^{n-1}.z^p$

∂U/∂z     $= px^{m}.y^n.z^{p-1}$

x + y + z = a

=> ∂U/∂x   =  ∂U/∂y  = ∂U/∂z =  ∂a/∂x    

$mx^{m-1}.y^n.z^p$ $= nx^{m}.y^{n-1}.z^p$$= px^{m}.y^n.z^{p-1}$

Dividing by $x^{m-1}.y^{n-1}.z^{p-1}$

=> myz = nxz  = pxy

my = nx  ,  mz = px ,   nz = py

x + y + z = a

x + nx/m  + px/m = a  

=> mx + nx + px = am

=> x = am /(m + n + p)

Similarly :

x = am/(m + n + p)

y = an/(m + n + p)

z = ap/(m + n + p)

Substitute these values in $x^m.y^n.z^p$    we get maximum value as

$(\dfrac{a}{m+n+p})^{m+n+p}. m^m.n^n.p^p$

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Answer 2

Answer:

(a/m+n+p)^m+n+p•m^mn^np^p