Question

Care este suma parametrilor reali a, b pentru care functia f: R->R, f(x)=$\sqrt[3]{ax^3+bx^2}$ admite ca asimptota oblica la +∞ dreapta y=x+1/3?

Answer 1

$\Big(x+\dfrac{1}{3}\Big)^3 = x^3+\dfrac{1}{27}+x^2+\dfrac{x}{3} = \\ \\ = x^3+x^2+\dfrac{x}{3}+\dfrac{1}{27} \\ \\ \Big(\sqrt[3]{x^3+x^2}\Big)^3 \simeq \Big(x+\dfrac{1}{3}\Big)^3,\quad \text{cand }x\to +\infty \\ \\ \Rightarrow \sqrt[3]{x^3+x^2}\approx x+\dfrac{1}{3},\quad \text{ cand }x\to +\infty \\ \\ \Rightarrow \Big(x+\dfrac{1}{3}\Big) \to \text{asimptota oblica spre }+\infty \text{ pentru } \sqrt[3]{x^3+x^2} \\ \\ \\\Rightarrow a =1;\quad b = 1\\ \\ \Rightarrow \boxed{S = 2}$