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Mathematics-Online course: Prepcourse Mathematics - Analysis - Extrema and Curve Sketching

Curve Sketching

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In order to qualitatively judge a function the following properties can be investigated:
\includegraphics[width=14.5cm]{Kurvendiskussion.eps}
(Authors: Höllig/Hörner/Abele)
Below one examines the function ALT=.

Symmetry: ALT= and ALT= being even, the function is even as well, that is ALT= is symmetric to the ALT=-axis.

Periodicity: The function isn't periodic.

Points of Discontinuity: The function is the composition of continuous functions and therefore has no points of discontinuity.

But as for the absolute value function, all derivatives are discontinuous at the origin.

Zeros: The exponential function is strictly positive, the absolute value function is nonnegative; thus there are no zeros.

Extrema:

ALT= ALT= ALT=  
  ALT= ALT=  

In addition, one has to examine the critical point ALT=, where the derivative is discontinuous. Because the domain is ALT=, one has to consider the behaviour of the function for ALT= to ALT=.

Since ALT= for ALT=, ALT= has at least one global minimum, yet no global maximum. Comparing the ALT=-values of critical points

ALT=

one finds that the global minimum lies at ALT=. ALT= having both a minimum and a maximum on ALT=, ALT= is a local maximum.

Inflection Points:

ALT= ALT= ALT=  
  ALT= ALT=  
  ALT= ALT=  

The third derivative doesn't vanish at these points.

Poles: The function has no poles.

Asymptotes: Since ALT= converges faster towards zero than ALT= for ALT= , ALT= has the asymptotes ALT= and ALT=.

ALT=
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  automatically generated 9/18/2007