I know that the range of $f(x)= ax^3 + bx^2 + cx + d$ is $\mathbb{R}$ but I can't prove it . It's obvious from the shape but I'm looking for a rigorous proof . I've tried to use IVT but didn't get any result .
2026-03-30 03:53:52.1774842832
Range of cubic equation
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HINT: write $$ax^3+bx^2+cx+d=x^3\left(a+\frac{b}{x}+\frac{c}{x^2}+\frac{d}{x^3}\right)$$ and compute the Limit for $x$ tends to $$\pm\infty$$ since $$\frac{b}{x},\frac{c}{x^2},\frac{d}{x^3}$$ tends to Zero for $$x$$ tends to$$\pm \infty$$ we get as a result $$\pm\infty \cdot \sign(a)$$