Can you solve?
I guess I'm tired of all the lame high school math questions on Steemit.
Here is a classical problem expected out of someone at the college level.
Assume a, b, and c are positive real numbers. Show:
I'll post the solution at the end of the week, after no one solves it. ;)
Regardless, I'll give an upvote to anyone who gives it the "ol' college try".
In order to simplify the expression, let’s set a + b = p and a+b+c = q
Then we have:
Combining the last two fractions:
If I rearrange the above slightly, I get:
If I continue the substitutions:
Now this contains all positive terms. The "greater or equal than" tells me I should attempt to find a minimum value, and to do that I need to introduce a negative term somehow. I do that by:
To make the numerator smaller, you want to maximize ab, which requires that a = b. Similarly, this also requires that c = b.
Therefore, (1/2)+(1/2)+(1/2) = 3/2 is the minimum value of the expression.
Good try so far!
a, b, & c are all independent variables. Note that a & b are (positive and) arbitrarily large, which means they are not necessarily maximized when they are equal. Indeed, for any choice of a, there are infinitely many b that make ab > a*a.
But here's a hint: You can make certain choices and not alter the independence of the 3 variables.
Thanks for at least trying! You made my week. :)
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