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talolard
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Homework Statement
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]
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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