How do you find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum?

Answer 1
Let's call the first number #x# Then the other will be #9-x# (as they add up to #9#
So the question translates to: Find the maximum of #y=x^2(9-x)=9x^2-x^3#
The maximum is when the derivative is set to #0#
#y'=18x-3x^2=3x*(6-x)=0->x=6# And the other number will be #9-6=3#
Check the answer: #6^2*3=108#
To be sure that this is a maximum you could check for: #5.9 harr 3.1# and for #6.1 harr 2.9#
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Answer 2

To find the two nonnegative numbers whose sum is 9 and maximize the product of one number and the square of the other, we use optimization techniques. Let's denote the two numbers as x and y, where x + y = 9. We need to maximize the function P(x, y) = x * y^2 subject to the constraint x + y = 9. We can solve this problem using the method of Lagrange multipliers or by expressing y in terms of x from the constraint equation and substituting it into the objective function to obtain a single-variable function. By finding the critical points and determining the maximum value, we can find the optimal values for x and y.

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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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