<s>[INST] Which number is greater, $11^{11}$ or $9^{12}$? 
Which number is greater than $11^{11}$ or $9^{12}$?

My work so far:
$11^{11}=285311670611>9^{12}=282429536481$.
But to verify the validity of equality should be in the range of easily verifiable calculations.
 [/INST] More generally, you are looking to prove: $x^x > (x-2)^{x+1}$. This can be done by taking log both sides, and its easier. Consider $f(x) = x\ln x - (x+1)\ln (x-2)$ on $(11, \infty)$, and taking log we have: $f'(x) = \ln x + 1 - \ln(x-2) - \dfrac{x+1}{x-2}$. We have $f''(x) = \dfrac{1}{x} - \dfrac{1}{x-2}+ \dfrac{3}{(x-2)^2}= \dfrac{(x-2)^2-x(x-2) + 3}{x(x-2)^2}= \dfrac{7-2x}{x(x-2)^2} < 0\implies f'(x) > f'(\infty) = 0 \implies f(x) > f(11) $
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