What Is Bigger $100^{100}$or $\sqrt{99^{99} \cdot 101^{101}}$

Question

Hello every what is bigger $100^{100}$or $\sqrt{99^{99} \cdot 101^{101}}$?

I tried to square up and I got $100^{200}$ or $99^{99} \cdot 101^{101}$ and I don't have an idea how to continue.

Answer 1

Taking logarithms, we see that we want to compare $f(100)$ and $\frac12(f(99)+f(101))$, where $f(x) = x\log x$. But $f(x)$ is a convex function (its second derivative $\frac1x$ is always positive), which means that the the secant line through $(99,f(99))$ and $(101,f(101))$ lies above the graph of the function. In particular, the fact that the midpoint of this secant line lies above the point $(100,f(100))$ on the graph is exactly the statement that $\frac12(f(99)+f(101)) > f(100)$, and so $\sqrt{99^{99}101^{101}} > 100^{100}$.