Mathematik-Online lexicon: Curve Sketching of a Rational Function

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	Mathematik-Online lexicon:
	Curve Sketching of a Rational Function

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overview

We analyze the function

$\displaystyle f(x)=\frac{5x^3+4x}{x^2-1} \,.$

Symmetry: The numerator is even and the denominator is odd. Thus the function is odd, i.e., symmetric with respect to the origin.

Periodicity: The function is not periodic.

Points of Discontinuity and Poles: The denominator of has simple zeros at $x=\pm 1$ . Since the numerator is nonzero at these points, the singularities are not removable and hence and are simple poles.

Zeros: The nominator vanishes at .

Extrema: The derivative

$\displaystyle f^\prime(x) = \frac{5x^4-19x^2-4}{(x^2-1)^2} = \frac{(x^2-4)(5x^2+1)}{(x^2-1)^2}$

vanishes at $x = \pm 2$ . The type of these critical points can be determined from the qualitative behavior of

. First we observe that

does not possess global extrema in view of the simple poles where the sign changes. Moreover, since $f(x)\to-\infty$ for $x\to -\infty$ and $x\to -1$ , the interval $\left(-\infty,-1 \right)$ must contain a local maximum. Similarly, $\left( 1, \infty \right)$ contains at least one local minimum. Hence, the zeros of $f^\prime$ both correspond to local extrema, a local maximum at