1. Determine if each of the following statements is true or false. Provide a cou

Question

1. Determine if each of the following statements is true or false. Provide a

counterexample for statements that are false and provide a complete proof

for those that are true.

(a) For all real numbers x and y

sqrt(xy) is less than or equal to (x+y)/2

This is obviously false as negative numbers are a counter example

.

(b) For all real numbers x and y

xy is less than or equal to ((x+y)/2)^2

So this is obviously true...not sure how to prove

c. For all nonegative real number x and y sqrt(xy) is less than or equal to (x+y)/2

This is true since it is the same as part 1 without negative numbers.....not sure how to prove

2)Use one of the true inequalities in Exercise (1) to prove the following proposition.

For each real number a, the value of x that gives the maximum value

of y = x (a -x) is x = (a/2)

How do I use the statements from part 1?

Explanation / Answer

ANSWER

follow this

Well, let's assume the opposite is true.

sqrt(xy) > (x + y) /2

If you were to square both sides you would get

xy > (x^2 + 2xy + y^2)/4

4xy > x^2 + 2xy + y^2

2xy > x^2 + y^2

x^2 - 2xy + y^2 < 0

(x - y)^2 < 0

Now assume that we take any two numbers x and y and subtract y from x. The answer is going to be some number, let's call it j. Now, either x is greater than y and j is positive, or x is less than y and j is negative. However, regardless, j^2 is always positive. So (x - y)^2 cannot be less than zer

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