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Curve discussion rational functions

6 important questions on Curve discussion rational functions

How do you determine the extreme of a function? And how do you know whether it is a minimum or a maximum?

x-value of the extreme: f'(x)=0
Plug result from f'(x) into f''(x)

f''(x)>0 -> Minimum
f''(x)<0 -> Maximum

y-value is found by plugging result from f'(x)=0 into f(x)

When is there a discontinuity and when is there a pole?

Discontinuity:
p(x0)/q(x0) -> p(x0)=0; q(x0)=0

Pole:
p(x0)/q(x0) -> p(x0) not 0; q(x0)=0

How is monotony determined in curve discussion?

Monotony is determined based on:

Extreme(s)
Zeros
Domain
Asymptote(s)

When is there an inflection point and when is there a saddle point?

Inflection point when:
f''(x0)=0
f'''(x0) different than zero

Saddle point:

f''(x0)=0
f'''(x0) different than zero
f'(x0)=0

How do you determine horizontal denominator?

Degree denominator > degree numerator
-> Then asymptote y=0

or

Degree denominator = degree numerator
-> Divide coefficients of the variables

How do you determine the oblique asymptote?

Assumption: The bigger the number plugged into the term, the smaller the outcome -> This term is left out, the rest is the asymptote.

Degree numerator > Degree denominator
-> Polynomial division

The question on the page originate from the summary of the following study material:

Mathematics

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