Question

Punctele b) şi c)
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Answer 1

$b) \text{ Ecuatia tangentei intr-un punct al graficului este :}\\\boxed{\boxed{\boxed{\bold{y-f(x_0)=f'(x_0)(x-x_0)}}}} \\\text{(am pus-o in trei chenare ca sa o tii minte)} .\text{In cazul de fata }x_0=1\\\text{Inlocuim:}\\y-f(1)=f'(1)(x-1)\\y-\dfrac{2}{e}= \dfrac{4}{e}(x-1) |\cdot e\\ey-2=4(x-1)\\ey-2=4x-4\\ey-4x+2=0$

$c) \text{Aici vom avea nevoie de tabel:}\\f'(x)=0\Leftrightarrow -x^2+2x+3=0\\\Delta=4+12=16\Rightarrow \sqrt{\Delta}=4\\x_1=\dfrac{-2+4}{-2}=-1\\x_2=\dfrac{-2-4}{-2}=3\\x~~~~~ |~~-1 ~~~~~~~~~~~3~~~~~~~~+\infty\\f'(x)| 0 ~~~~~+++~~0~~---~\\f(x)~|  f(-1) ~~    \nearrow ~~f(3)~~\searrow ~~~0 \\\text{Faci tu un tabel mai frumos in caiet :) }.\\\text{De pe tabel se observa ca } f(-1)\leqslant x\leqslant f(3)~ \forall x\in[-1,\infty)\\\text{Inlocuind ,se obtine concluzia.}$