solve the following equation :
$$3^x +4^x = 5^x \text{ in } \Bbb R$$
the trivial answer is $x=2$ .
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sooo... i guess you solved the equation? – 5xum Mar 25 '16 at 08:19
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5Interestingly, $2$ is the only real solution to this equation – learner Mar 25 '16 at 08:23
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@5xum. no i can not yet – amir bahadory Mar 25 '16 at 08:27
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فرانك جواب خوبي داده در پايين. واقعا عالي حل كرده. پيكير همين روش باش – Mikasa Mar 25 '16 at 08:28
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@amirbahadory But you did. You found a solution, right? – 5xum Mar 25 '16 at 08:29
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@5xum.yes i find x=2 – amir bahadory Mar 25 '16 at 08:31
2 Answers
9
Let $f(x)=\frac{3^x+4^x}{5^x}=(\frac{3}{5})^x+(\frac{4}{5})^x$, not hard to see $f$ is continuous and monotonically decreasing so $f(x)=1$ only has unique solution $x=2$.
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$$\left( \frac{3}{5} \right)^x+\left( \frac{4}{5} \right)^x=1$$ Then $x=2$
$f(x)=\left( \frac{3}{5} \right)^x+\left( \frac{4}{5} \right)^x$decreasing function
Roman83
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