Question

Ajutați ma va rog Lab subiectul3 ex1

Answer 1

Răspuns

Explicație pas cu pas:

$\displaystyle f:\mathbb{R}\rightarrow \mathbb{R},f(x)=x^3-12x+5\\a)\limits\lim_{x\to \infty} \dfrac{f(x)}{2x^3+5}=\limits\lim_{x\to \infty}\dfrac{x^3-12x+5}{2x^3+5}=\limits\lim_{x\to \infty}\dfrac{x^3\left(1-\dfrac{12}{x^2}+\dfrac{5}{x^3}\right)}{x^3\left(2+\dfrac{5}{x^3}\right)}=\bold{\dfrac{1}{2}}\\b)f'(x)=3x^2-12\\f'(x)=0\Rightarrow 3x^2-12=0\\3x^2=12\\x^2=4\\|x|=2\\x\in \{\pm 2\}$

$\displaystyle c)Solutiile~ecuatiei~f'(x)=0~sunt~\pm 2\\\limits \lim_{x\to\infty} f(x)=\infty ,\lim_{x\to-\infty} f(x)=-\infty\\Faci~un~tabel~tinand~cont~ca~f~este \nearrow pe (-\infty,-2)\cup [2,\infty)~si~\\descrescatoare~pe~[-2,2). Prin~urmare~x=2~si~x=-2~sunt~puncte\\de~extrem~local.\\f(-2)=21 \Rightarrow A(-2,21)\\f(2)=-17 \Rightarrow B(2,-17)$