Matematické Fórum

stuart clark

If $f$ is a double differentiable function and satisfy the condition

$f(0)=0$ and $f(1)=0$ and $\frac{d^2}{dx^2}\left(e^{-x}f(x)-x^2\right)>0\;\forall x\in (0,1)$

Then number of values of $x$ for which $f(x)-3=(x^2-x)e^x\;\forall x\in (0,1)$
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I have tried like this way\;, Given $\frac{d^2}{dx^2}\left(e^{-x}f(x)-x^2\right)>0$

So $\int\frac{d^2}{dx^2}\left(e^{-x}f(x)-x^2\right)dx>\int 0\cdot dx$

So $\frac{d}{dx}\left(e^{-x}f(x)-x^2\right)>c\Rightarrow \int\frac{d}{dx}\left(e^{-x}f(x)-x^2\right)dx>\int cdx$

So $f(x)>e^{x}x^2+cxe^x+de^x$

Now How can i solve it after that, Thanks

Brano · 17. 03. 2016 10:04

consider the function
$g(x)=e^{-x}f(x)-x^2+x$
then $g(0)=g(1)=0$ and $g''(x)>0$ i.e. $g$ is convex and therefore for every $x\in(0,1)$ we have $g(x)<0$
now you are asking for which $x\in(0,1)$ holds $g(x)=3e^{-x}>0$ - the answer is for none.

stuart clark · 20. 03. 2016 11:46

Thanks ↑ Brano:.

Matematické Fórum

#1 16. 03. 2016 12:05 — Editoval stuart clark (16. 03. 2016 12:07)

values of $x$ in function equation

#2 17. 03. 2016 10:04

Re: values of $x$ in function equation

#3 20. 03. 2016 11:46

Re: values of $x$ in function equation

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