Maximum of function f(x,y,z)=log(x)+log(y)+3log(z) for x,y,z>0, x^2+y^2+z^2=5r^2

[danode]

Hi!

I am having struggles solving this following task:

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Find the maximum for the function:

. . . . .\(\displaystyle f(x,\, y,\, z)\, =\, \log(x)\, +\, \log(y)\, +\, 3\log(z)\)

...under constraints:

. . . . .\(\displaystyle x,\, y,\, z\, >\, 0,\, x^2\, +\, y^2\, +\, z^2\, =\, 5r^2,\)

...where r is given. Use this to show the differences:

. . . . .\(\displaystyle abc^3\, \leq\, 27\, \left(\dfrac{a\, +\, b\, +\, c}{5}\right)^5,\)

...for all positive numbers a, b, and c.

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Could anyone be so kind to help me with this? This task is a bit beyond my knowledge.

 

[JeffM]

Hi!

I am having struggles solving this following task:

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Find the maximum for the function:

. . . . .\(\displaystyle f(x,\, y,\, z)\, =\, \log(x)\, +\, \log(y)\, +\, 3\log(z)\)

...under constraints:

. . . . .\(\displaystyle x,\, y,\, z\, >\, 0,\, x^2\, +\, y^2\, +\, z^2\, =\, 5r^2,\)

...where r is given. Use this to show the differences:

. . . . .\(\displaystyle abc^3\, \leq\, 27\, \left(\dfrac{a\, +\, b\, +\, c}{5}\right)^5,\)

...for all positive numbers a, b, and c.

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Could anyone be so kind to help me with this? This task is a bit beyond my knowledge.

I must admit that I do not have a clue what "find the differences" means. Perhaps it is clear what that means in the context of ALL the tasks, but you will need to clarify.

WE have no idea what you know so it is impossible to give an answer that we know is not "beyond your knowledge."

Let's try this.

Do you know how to find local extrema of a multivariate differentiable function?

If you do, have you been taught about LaGrangian multipliers?

If so, how would you set up the relevant function to maximize?