Find three positive numbers whose sum is 140 and whose product is a maximum. (Enter your answers as a comma-separated list.)

Step-by-step explanation:

Let the numbers be

Such that:

Make z the subject

For their product to be maximum, we have:

Substitute in

Open bracket

Differentiate w.r.t x and y

Since the products are maximum, then

For

Factorize:

Split

Make y the subject

For

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Substitute y = 0

Factorize

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Substitute

Re-arrange

Factor x out

Divide through by x

Recall that:

Take LCM

Recall that:

Take LCM

Hence, the numbers are: