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Question Number 183535 by HeferH last updated on 26/Dec/22

$$Whoisgreater?{70}^{71}or{71}^{70}$$

Answered by Frix last updated on 26/Dec/22

$$1^2<2^1$$$$2^3<3^2$$$$===$$$$3^4>4^3$$$$4^5>5^4$$$$...$$$$n^{n+1}>{(n+1)}^n\mathrm{\forall }n⩾3$$

Commented by HeferH last updated on 26/Dec/22

$$thanks:)$$

Answered by mr W last updated on 26/Dec/22

$$f\left(x\right)=x^{\frac{1}{x}}=e^{\frac{\ln x}{x}}$$$$f^{\prime }\left(x\right)=x^{\frac{1}{x}}\left(\frac{1-\ln x}{x^2}\right)$$$$weseeforx>e,f^{\prime }\left(x\right)<0,thatmeans$$$$f\left(x\right)isstrictlydecreasingforx>e,$$$$i.e.3^{\frac{1}{3}}>4^{\frac{1}{4}}>5^{\frac{1}{5}}>...$$$${70}^{\frac{1}{70}}>{71}^{\frac{1}{71}}$$$${70}^{\frac{71}{70}}>71$$$$\Rightarrow {70}^{71}>{71}^{70}$$$$$$$$similarlyyoucanalsogete.g.$$$${100}^{200}>{200}^{100}$$

Commented by HeferH last updated on 26/Dec/22

$$thanks!$$

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