a) [tex]f'(x)=e^x-1[/tex]
b) [tex]f'(x)=0\implies e^x=1\implies x=0[/tex]
[tex]x < 0\implies e^x < 1\implies f'(x) < 0[/tex]
[tex]\implies\boxed{ f \text{ descrescatoare pe }(-\infty, 0)}[/tex]
[tex]\text{Analog }\boxed{f\text{ crescatoare pe }(0,\infty)}[/tex]
[tex]\text{iar }\boxed{x=0}\text{ este punct de minim}[/tex]
c) [tex]e^{x^2}+e^x\ge x^2+x+2\iff e^{x^2}+e^x-x^2-x-2\ge0[/tex]
se observa ca [tex]f(x^2)+f(x)=e^{x^2}-x^2-1+e^x-x-1=e^{x^2}+e^x-x^2-x-2[/tex]
dar stim de mai devreme ca [tex]x=0[/tex] este punct de minim pentru [tex]f[/tex]
inseamna ca [tex]f(x)\ge f(0) \; \forall x\in\mathbb{R}[/tex]
dar [tex]f(0)=e^0-0-1=1-1=0[/tex]
deci [tex]f(x)\ge 0 \; \forall x\in\mathbb{R}[/tex]
[tex]\implies f(x^2)+f(x)\ge0\ \forall x\in\mathbb{R}[/tex]
[tex]\implies e^{x^2}+e^x-x^2-x-2\ge0\ \forall x\in\mathbb{R}[/tex]
[tex]\implies\boxed{e^{x^2}+e^x\ge x^2+x+2\ \forall x\in\mathbb{R}}[/tex]
d) [tex]f''(x)=e^x\ge0\ \forall x\in\mathbb{R}\implies \boxed{f \text{ convexa}}[/tex]
e) limita aceea este tocmai definitia derivatei lui f in punctul x=1, deci
[tex]\displaystyle \lim_{x\to1}\cfrac{f(x)-f(1)}{x-1}=f'(1)=e^1-1=\boxed{e-1}[/tex]
f) ecuatia tangentei la grafic in punctul A(0, 0) este:
[tex]\displaystyle y-f(0)=f'(0)(x-0)[/tex]
dar [tex]f(0)=f'(0)=0[/tex], deci ecuatia tangentei este
[tex]\boxed{y=0}[/tex]
