Which is bigger: or ?
We know that .
Let us conjecture that .
This means that , that is, . Therefore, if our conjecture is true, we must have .
Let us consider the function . Find its minimum on . You will find that the minimum of is , and this minimum is reached when .
But we know that , thus the minimum is never reached with .
. You can follow the steps backwards to finally prove that
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That was the nearly-complete proof. Now, you only have to show that the minimum of is when , and you are done
Lol, I wasn't sure my proof was going to pass since this is the algebra forum and my proof is already half-way into calculus, but Drex actually brought in the integral !
Nice proof though, everything is already trivially proved and one just has to put the bits together
graphing actually occured to me, but i don't like this since actually graphing that graph without the use of calculus would be a pain. if you asked a pre-university student to do this, they'd probably just use a graphing utility or plot points with a calculator, in which case they wouldn't need the graph to find the answer to the problem, they'd just use the calculator. i think the idea here is to manually do the problem.