The sum of one number and the square of another number is #27#. What is the maximum product of these two numbers, and what two numbers give this product?

Answer 1

#3#. The maximum product is #54# and the numbers that will give this maximum are #18# and #3#.

Here is the answer to #3. I'll post my answers one at a time.

Let the numbers be #x# and #y#.
#y + x^2 = 27#
#y = 27 - x^2#
Call a new function #P#, where #P = xy#.
#P = x(27 - x^2)#
#P = 27x - x^3#

This has a derivative of

#P' = -3x^2 + 27#
The critical numbers here will occur when #P' = 0#.
#0 = -3x^2 + 27#
#0 = -3(x^2 - 9)#
#0 = -3(x + 3)(x - 3)#
#x = +- 3#
Now let's see which one is our maximum. At #x = 1#, the derivative is increasing because #P'(1) = -3(1)^2 + 27(1) = 24#, which is positive. Therefore, #x = +3# will be a maximum.
The numbers will hence be #y = 27 - 3^2 = 18#. The product will be #54#. A graphical verification yields similar results (aka the absolute maximum on #x >0# is at #(3, 54)#, which means that an #x# value of #3# gives a maximum product of #54#. ).

graph{27x - x^3 [-99.95, 87.5, -16.8, 76.95]}

Hopefully this helps!

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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