Question

exercitiul 1b va rog, nu mi pot da seama cum aplic proprietatile ln si ale puterilor ​

Answer 1

$f(x) = e^{x\cdot \ln x}\\ \\ f'(x) = e^{x\cdot \ln x}\cdot ( \ln x +1)\\ \\ f'(x) = 0 \Rightarrow e^{x\cdot \ln x} = 0,\quad (F)\,\,\text{(deoarece o exponentiala nu poate fi 0)} \\ \\ \ln x+1 = 0\Rightarrow \ln x = -1 \Rightarrow x = e^{-1}\\\\\lim\limits_{x\searrow 0} f'(x) = \lim\limits_{x\searrow 0} e^{\ln x^x}\cdot (\ln x+1) = e^{\ln 1}\cdot (-\infty +1) = -\infty < 0\\ \\ \Rightarrow f'(x) < 0,\quad x< e^{-1} \\ \Rightarrow f'(x) > 0,\quad x > e^{-1}$

$\Rightarrow f(x) \text{ strict descrescatoare cand } x < e^{-1} \\ \Rightarrow f(x) \text{ strict crescatoare cand } x > e^{-1} \\ \\ \Rightarrow f_{min} = f(e^{-1}) =e^{e^{-1}\cdot \ln e^{-1}} =e^{e^{-1}\cdot (-\ln e)} = e^{e^{-1}\cdot (-1)} = \\ \\ =e^{-\frac{1}{e}}\Rightarrow \boxed{f_{min}=\dfrac{1}{e^{\frac{1}{e}}}}}$