Cfrgtkky

3 1 $begingroup$ Let $f(x)$ be a twice differentiable function on the interval $(-infty,+infty)$ and $f''(x)>f(x)$ for all $x$ . Prove that $f$ has at most two zero points. I'm trying to prove it by contradiction,but I can't work it out. real-analysis calculus share | cite | improve this question edited Mar 9 at 19:17 Prem 113 4 asked Mar 9 at 14:05 Maxwell Maxwell 44 5 $endgroup

3 $begingroup$ Numbers $1,2,ldots,n$ are arranged into a circle. What is the maximum product of the differences $|x_1-x_2|times|x_2-x_3|timescdotstimes|x_{n-1}-x_n|times|x_n-x_1|$? I think the maximum should occur when the numbers are arranged $n,1,n-1,2,n-2,3,ldots$. The sum for this arrangement is $(n-1)(n-2)cdots1cdotlfloor n/2rfloor = (n-1)!cdotlfloor n/2rfloor$. This question was inspired by the question asking for the maximum sum. There, it is possible to prove optimality by noting that we have $2n$ terms ($n$ with $+$ and $n$ with $-$), and each number occurs twice. Here, it is still true that we have $2n$ terms ($n$ with $+$ and $n$ with $-$), and each number occurs twice. But since we're taking the product instead of the sum, optimality is no longer clear.