Question

the range of function, f(x)=x^3-1/x-1

Answer 1

Answer on Question #44654 - Math - Functional Analysis

Find the range of function

$f(x)=x^{3}-\frac{1}{x}-1$

Firstly, let’s find the derivative.

$f^{\prime}(x)=3x^{2}+\frac{1}{x^{2}}\geq 2\sqrt{3},$

due to Cauchy inequality. Hence, function is increasing on $(-\infty,0)$ and $(0,+\infty)$. Let’s consider the second case.

$\lim_{x\to 0+}f(x)=\lim_{x\to 0+}(x^{3}-\frac{1}{x}-1)=-\infty,$

due to the fact that $x^{3}$ and $-1$ are continuous in .

$\lim_{x\to+\infty}f(x)=\lim_{x\to+\infty}(x^{3}-\frac{1}{x}-1)=+\infty,$

due to the fact that $-\frac{1}{x}\to 0(x\to\infty)$. Because of the fact that

$f(x)=x^{3}-\frac{1}{x}-1$

is continuous on $(0,+\infty)$, it reaches every value from lower to upper bound, therefore, it reaches every value in $(-\infty,+\infty)$

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