Question

Ma puteti ajuta va rog . Trebuie sa demonstrez ca e la x -e la -x este crescatoare . Va rog

Answer 1

[tex]\displaystyle\\ f(x)=x -e^{-x}\\\\ \text{Calculam derivata functiei.}\\\\ f'(x)=\Big(x -e^{-x}\Big)' =\\\\ = x' - \Big(e^{-x}\Big)' = x'- (-x)'\cdot e^{-x} =1 -(-1)\cdot e^{-x} = 1+e^{-x} =\boxed{1+ \frac{1}{e^x}}\\\\ [/tex]


[tex]\displaystyle\\ \boxed{e^x \ \textgreater \ 0} ~\text{pentru oricare } ~x\in \mathbb{R}\\\\ \Longrightarrow~~~ \boxed{\frac{1}{e^x}\ \textgreater \ 0} ~\text{pentru oricare } ~x\in \mathbb{R}\\\\ \Longrightarrow~~~ \boxed{1+\frac{1}{e^x}\ \textgreater \ 0} ~\text{pentru oricare } ~x\in \mathbb{R}\\\\ \Longrightarrow~~~ \boxed{f'(x)}\ \textgreater \ 0} ~\text{pentru oricare } ~x\in \mathbb{R}\\\\ \Longrightarrow~~~\boxed{f(x)=x -e^{-x} ~\text{este crescatoare.} } [/tex]