Can someone help me with math homewoHello! I have problems with my math homework assignments. Can someone help me please?

Exercise: 1) Write the number 60 in different ways as the sum of two numbers x + y. More general: With which summands formed in this way is the product x * y largest?

2) A number is subtracted from 13. The same number is added to 21. The results of both calculations are multiplied together. How to choose the number for the product to be the largest?

Thanks in advance! :)

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Question #1:We can use Lagrange Multipliers. The Lagrangian is:

\[ L = x y + \lambda (x + y - 60) \]

Take partial derivatives and set equal to zero:

\[\frac{\partial{L}}{\partial{x}} = y + \lambda = 0 \\ \frac{\partial{L}}{\partial{y}} = x + \lambda = 0 \\ \frac{\partial{L}}{\partial{\lambda}} = x + y - 60 = 0 \]

Solving yields:

\[x = 30 \\ y = 30 \\ \lambda = -30\]

And we know this is a maximum since \(30 \times 30 > 40 \times 20\)

Question #2:Maximize the following quantity:

\[Q = (13 - x)(21 + x)\]

Differentiate and set to zero:

\[(13 - x )(1) + (21 + x )(-1) = 0 \\ x = -4\]

We know that this is a maximum because putting in any other value for \(x\) yields a smaller number. Interestingly, this value for \(x\) makes both multiplied quantities equal to \(17\).

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Yes, it's can bee calculus!

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Can it be calculus, or does it have to be algebra?

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