(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let f be a function from R to R such that [tex] f'(x)>f(x), f(0)=1 [/tex] for every x. prove that for every [tex] x>0 f(x)>e^x [/tex]

3. The attempt at a solution

From the given information we know that [tex] f'(0) \geq 0 [/tex] and so [tex] f'(x) \geq 0 [/tex]

define [tex] h(x) = f(x)-e^x [/tex]

so h(0) =0

[tex] h'(x)=f'(x) -e^x = lim_{h ->0 } \frac {f(x+h)-f(x)}{h} -e^x= lim_{h ->0 } \frac {f(x+h)-f(x) -he^x}{h}= lim_{h ->0 } \frac {f(x+h)-f(x) -0}{h} = f'(x) \geq 0 [/tex]

Then h(x) is montonous rising and so is never smaller then zero. thus [tex] f(x) \geq e^x [/tex]

Is this correct?

Thanks

Tal

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# Homework Help: Calculous proof question

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