largest solution to 2^x=x^2

[RJMilazzo]

I found these 3 roots of the equation, 2^x=x^2, {2,4, -.766664...}

How can I prove that 4 is the largest root of this equation?

Thanks,

Angela

rjmi...@aol.com

[Denis Feldmann]

RJMilazzo wrote:
> I found these 3 roots of the equation, 2^x=x^2, {2,4, -.766664...}
>
> How can I prove that 4 is the largest root of this equation?

Try studying the function x->2^x-x^2, by differentiating twice

>
> Thanks,
>
> Angela
>
> rjmi...@aol.com

[William Elliot]

from rjmi...@aol.com
_ I found these 3 roots of the equation, 2^x=x^2, {2,4, -.766664...}
_ How can I prove that 4 is the largest root of this equation?
The old fashion way: e < 4; sqr e < 2; 1/2 < log 2
f(x) = 2^x - x^2; f'(x) = (log 2)2^x - 2x
f(4) = 0; f'(x) > 2^(x-1) - 2x >= 0 for all x >= 4

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[Jaime Gaspar]

"RJMilazzo" <rjmi...@aol.com> escreveu na mensagem
news:20020817060044...@mb-fw.aol.com...

> I found these 3 roots of the equation, 2^x=x^2, {2,4, -.766664...}
>
> How can I prove that 4 is the largest root of this equation?

1. Show that x = 4 is a root of the equation 2^x = x^2;

2. Show that for all x > 4, 2^x > x^2, therefore there aren't solutions of
the equation 2^x = x^2. You may prove this showing that for all x > 4, f(x)
= 2^x - x^2 > 0. This last one can be done by showing that f is continuous
(is enough to show that for x >= 4) and that f'(x) = 2^x ln 2 - 2x > 0 (for
all x > 4);

Regards,

Jaime Gaspar
______________________________
Homepage: www.jaimegaspar.com
E-mail: e-m...@jaimegaspar.com

[Robert G. Wilson v]

Or more precisely

-0.7666646959621230931112044225103148480066753466698320584608843769355527957248724228530292096979025305654798600517113341669299\
8689443261283852599524956842765567011510473097008646045078795354838672851703083164522897799553624414367241313195901538874827983\
9666603963077476414680764013655596113811484266857096975519043100272114308642265825970771064427251608468427527418776193259409359\
0226244100825762231174877946916522089409297748008681203161543896041713201591343927432115141261743664864515466575485955256411587\
5769352493985757358334381897982679466060258852612302153112505990272658836822097034017667023316797362128634357825438269222114031\
6041495244249750646191706066900175194897417287910968234879521824683525752516110916068039128093041717515160318994457749295745303\
0578105797544005159931045350219873956458569679085952446539726125958616496858147175926560965375034239874926482343865205707082367\
8168693858766435145256956438491135488492460454427546206086685951184150072151679705909085158930951527567011392686592642916102332\
54873