Christian K.

# There are infinitely many pairs of numbers such that the greater one is 3 more than 2 times the lesser one.

(Can you list some such pairs?) Find the pair whose product is as small as possible.

## 2 Answers By Expert Tutors

By:

Mathematics Teacher - NCLB Highly Qualified

PhD in Engineering with 20 yrs of Math and Science Teaching Experience

Mark M.

tutor
I understand your work if the numbers are assumed to be positive. What of the pair (1, -1) whose product is -1?
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09/11/15

Alexander B.

tutor
It may be, indeed, a valid suggestion. Best regards,
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09/11/15

Mark M.

tutor
It is more than a suggestion. It conforms to the requirements of the problem and is less than the solution that you provided.
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09/11/15

Mark M.

tutor
Rewriting the given: y = (x - 3)/2.
For x > 3, y > 0, and xy > 0
For 0 < x < 3, y < 0, and xy < 0
For x < 0, y < 0, and xy > 0.

Since negative numbers are less than positive numbers the possible x values are such: 0 < x < 3.
I am still pondering how to determine the minimal value of xy.
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09/11/15

Alexander B.

tutor
I've updated my answer for both positive and negative values. Best regards,
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09/12/15

Alexander B.

tutor
My answer includes a theoretical proof that a pair of numbers [1.5, -0.75] (i.e. x=1.5; y=-0.75) corresponds to the smallest product of xy=-1.125 (re: finding quadratic extrema). Best regards,
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09/12/15

Mark M.

tutor
Formulas are the result of mathematical thinking. They do not replace mathematical thinking.
I am encouraged that my arithmetic process/thinking is verified by differential calculus.
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09/12/15

Alexander B.

tutor
Yeah, that's right. I've added a simplified explanation without reference to the calculus pertinent to this particular case, i.e. well-known properties of the quadratic parabola. But at college math level (as indicated) those folks probably are already familiar with more universal approach to the functional analysis using basic calculus. Best regards,
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09/12/15

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