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Who-is-greater-70-71-or-71-70-




Question Number 183535 by HeferH last updated on 26/Dec/22
Who is greater? 70^(71)  or  71^(70)
$${Who}\:{is}\:{greater}?\:\mathrm{70}^{\mathrm{71}} \:{or}\:\:\mathrm{71}^{\mathrm{70}} \\ $$
Answered by Frix last updated on 26/Dec/22
1^2 <2^1   2^3 <3^2   ===  3^4 >4^3   4^5 >5^4   ...  n^(n+1) >(n+1)^n ∀n≥3
$$\mathrm{1}^{\mathrm{2}} <\mathrm{2}^{\mathrm{1}} \\ $$$$\mathrm{2}^{\mathrm{3}} <\mathrm{3}^{\mathrm{2}} \\ $$$$=== \\ $$$$\mathrm{3}^{\mathrm{4}} >\mathrm{4}^{\mathrm{3}} \\ $$$$\mathrm{4}^{\mathrm{5}} >\mathrm{5}^{\mathrm{4}} \\ $$$$… \\ $$$${n}^{{n}+\mathrm{1}} >\left({n}+\mathrm{1}\right)^{{n}} \forall{n}\geqslant\mathrm{3} \\ $$
Commented by HeferH last updated on 26/Dec/22
thanks :)
$$\left.{thanks}\::\right) \\ $$
Answered by mr W last updated on 26/Dec/22
f(x)=x^(1/x) =e^((ln x)/x)   f′(x)=x^(1/x) (((1−ln x)/x^2 ))  we see for x>e, f′(x)<0, that means  f(x) is strictly decreasing for x>e,  i.e. 3^(1/3) >4^(1/4) >5^(1/5) >...  70^(1/(70)) >71^(1/(71))   70^((71)/(70)) >71  ⇒70^(71) >71^(70)     similarly you can also get e.g.  100^(200) >200^(100)
$${f}\left({x}\right)={x}^{\frac{\mathrm{1}}{{x}}} ={e}^{\frac{\mathrm{ln}\:{x}}{{x}}} \\ $$$${f}'\left({x}\right)={x}^{\frac{\mathrm{1}}{{x}}} \left(\frac{\mathrm{1}−\mathrm{ln}\:{x}}{{x}^{\mathrm{2}} }\right) \\ $$$${we}\:{see}\:{for}\:{x}>{e},\:{f}'\left({x}\right)<\mathrm{0},\:{that}\:{means} \\ $$$${f}\left({x}\right)\:{is}\:{strictly}\:{decreasing}\:{for}\:{x}>{e}, \\ $$$${i}.{e}.\:\mathrm{3}^{\frac{\mathrm{1}}{\mathrm{3}}} >\mathrm{4}^{\frac{\mathrm{1}}{\mathrm{4}}} >\mathrm{5}^{\frac{\mathrm{1}}{\mathrm{5}}} >… \\ $$$$\mathrm{70}^{\frac{\mathrm{1}}{\mathrm{70}}} >\mathrm{71}^{\frac{\mathrm{1}}{\mathrm{71}}} \\ $$$$\mathrm{70}^{\frac{\mathrm{71}}{\mathrm{70}}} >\mathrm{71} \\ $$$$\Rightarrow\mathrm{70}^{\mathrm{71}} >\mathrm{71}^{\mathrm{70}} \\ $$$$ \\ $$$${similarly}\:{you}\:{can}\:{also}\:{get}\:{e}.{g}. \\ $$$$\mathrm{100}^{\mathrm{200}} >\mathrm{200}^{\mathrm{100}} \\ $$
Commented by HeferH last updated on 26/Dec/22
thanks !
$${thanks}\:! \\ $$

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