Question

f:{1 ∞)->R f|(x)=$e^{x}-\frac{1}{x}$ .Aratati ca f(x)>0 ∀x$\geq 1$

Answer 1

$x\geq 1 \Rightarrow \dfrac{1}{x}\leq 1 \Rightarrow -\dfrac{1}{x}\geq -1\quad (*)\\ \\ e^x \text{ functie strict crescatoare pe }\mathbb{R}. \\ \\ x\geq 1 \Rightarrow e^x\geq e^1 \Rightarrow e^x \geq e\quad (**) \\ \\ \text{Adunam cele doua inegalitati:} \\ \\ \Rightarrow e^x+\Big(-\dfrac{1}{x}\Big) \geq e+(-1)\\ \\ \Rightarrow f(x) \geq e-1 >0 \\ \\ \Rightarrow f(x)>0,\quad \forall x\geq 1$