Question

Trace the curve y=
$\sqrt[3]{ {x2 - 1 } }$
, and state all the properties you use to trace it.​

Answer 1

Answer:

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Answer 2

Trace the curve y=  $\sqrt[3]{x^2-1}$ , and state all the properties you use to trace it.​

$\mathrm{Domain\:of\:}\:\sqrt[3]{x^2-1}\::\quad \begin{bmatrix}\mathrm{Solution:}\:&\:-\infty \:<x<\infty \\ \:\mathrm{Interval\:Notation:}&\:\left(-\infty \:,\:\infty \:\right)\end{bmatrix}\\\\\mathrm{Range\:of\:}\sqrt[3]{x^2-1}:\quad \begin{bmatrix}\mathrm{Solution:}\:&\:f\left(x\right)\ge \:-1\:\\ \:\mathrm{Interval\:Notation:}&\:[-1,\:\infty \:)\end{bmatrix}\\\\\mathrm{Inverse\:of}\:\sqrt[3]{x^2-1}:\quad \sqrt{x^3+1},\:-\sqrt{x^3+1}\\\\\mathrm{Domain\:of\:}\:\sqrt{x^3+1}\::\quad x\ge \:-1$

$\mathrm{Axis\:interception\:points\:of}\:\sqrt[3]{x^2-1}:\quad \mathrm{X\:Intercepts}:\:\left(1,\:0\right),\:\left(-1,\:0\right),\:\mathrm{Y\:Intercepts}:\:\left(0,\:-1\right)\\\\\mathrm{Extreme\:Points\:of}\:\sqrt[3]{x^2-1}:\quad \mathrm{Saddle}\left(-1,\:0\right),\:\mathrm{Minimum}\left(0,\:-1\right),\:\mathrm{Saddle}\left(1,\:0\right)$

$\mathrm{Asymptotes\:of}\:\sqrt[3]{x^2-1}:\quad \mathrm{None}$

The curve is symmetrical about the origin.

Monotonicity:

$\dfrac{d}{dx}\left(\sqrt[3]{x^2-1}\right)=\dfrac{2x}{3\left(x^2-1\right)^{\frac{2}{3}}}$

Since dy/dx > 0, the graph is monotonically increasing.

$\dfrac{d^2y}{dx^2}\left(\dfrac{2x}{3\left(x^2-1\right)^{\frac{2}{3}}}\right) =\dfrac{2\left(-x^2-3\right)}{9\left(x^2-1\right)^{\frac{5}{3}}}$

Since d²y/dx² ≠ 0 at x = 0

Therefore, the point of intersection is not origin.