Partial differentiation

[StephenR]

Hi, This is a question from Silvanus Thompson's Calculus Made Easy. Verify that the sum of three quantities x,y,z, whose product is a constant k, is maximum when these three quantities are equal. Any help appreciated!

 

[Deleted member 4993]

Hi, This is a question from Silvanus Thompson's Calculus Made Easy. Verify that the sum of three quantities x,y,z, whose product is a constant k, is maximum when these three quantities are equal. Any help appreciated!

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Please share your work/thoughts and context of the problem (what is the subject topic?) - so that we know where to begin to help you.

Hint: S = x + y + z ..........................This is a function of three variables (with a constraint k = x * y * z).

If it were function of single variable - what would you have done...... calculate dS/dx and set it to zero for maxima.

You have to do similar things here.....

 

[HallsofIvy]

You don't say at what "level" in the book this is but to me it cries out for "Lagrange multipliers". You want to maximize f(x,y,z)= x+ y+ z subject to the constraint g(x,y,z)= xyz= 5. The gradient of f is \(\displaystyle \nabla f= <1, 1, 1>\). The gradient of g is \(\displaystyle \nabla g= <yz, xz, xy>\). Where f is a maximum (or minimum), subject to g= 5, those two vectors must be parallel: \(\displaystyle <yz, xz, xy>= \lambda <1, 1, 1>\) for some constant \(\displaystyle \lambda\) (the "Lagrange multiplier"). That is we must have \(\displaystyle yz= \lambda\), \(\displaystyle xz= \lambda\), and \(\displaystyle xy= \lambda\). Since getting a value of \(\displaystyle \lambda\) is not necessary to solving this problem you might find it simplest to first eiminate \(\displaystyle \lambda\) by dividing one equation by another.