Concavity, asymptotes and more..

Oct 2009
8
0
f(x)= x^2/(x-2)^2

Find:
a) Vertical and horizontal asymptotes.
b) Intervals of increase or decrease.
c) local maximum and minimum values.
d) intervals of concavity and inflection points
e) Sketch the graph of f(x)
 

Prove It

MHF Helper
Aug 2008
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f(x)= x^2/(x-2)^2

Find:
a) Vertical and horizontal asymptotes.
b) Intervals of increase or decrease.
c) local maximum and minimum values.
d) intervals of concavity and inflection points
e) Sketch the graph of f(x)
a) Find the values that \(\displaystyle x\) and \(\displaystyle f(x)\) can not take.

b) Derivative \(\displaystyle > 0\) means increasing.
Derivative \(\displaystyle < 0\) means decreasing.

c) Derivative \(\displaystyle =0\) means a stationary point.

If the second derivative \(\displaystyle >0\) at that point, you have a minimum.

If the second derivative \(\displaystyle <0\) at that point, you have a maximum.

If the second derivative \(\displaystyle =0\) the test is inconclusive. Check the derivative at values near that point.

d) The function is concave when the second derivative \(\displaystyle < 0\).

The function is convex when the second derivative \(\displaystyle > 0\).

The function has inflection points when it goes from convex to concave or vice versa. I.e. where the second derivative \(\displaystyle = 0\).

e) Use the information found in a) - d).
 
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HallsofIvy

MHF Helper
Apr 2005
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a) Find the values that \(\displaystyle x\) and \(\displaystyle f(x)\) can not take.
A horizontal asymptote is NOT a value "f(x) cannot take". If the limit, as x goes to infinity or negative infinity, is a finite number, then y= that number is the horizontal aymptote.

b) Derivative \(\displaystyle > 0\) means increasing.
Derivative \(\displaystyle < 0\) means decreasing.

c) Derivative \(\displaystyle =0\) means a stationary point.

If the second derivative \(\displaystyle >0\) at that point, you have a minimum.

If the second derivative \(\displaystyle <0\) at that point, you have a maximum.

If the second derivative \(\displaystyle =0\) the test is inconclusive. Check the derivative at values near that point.

d) The function is concave when the second derivative \(\displaystyle < 0\).

The function is convex when the second derivative \(\displaystyle > 0\).

The function has inflection points when it goes from convex to concave or vice versa. I.e. where the second derivative \(\displaystyle = 0\).

e) Use the information found in a) - d).
 
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Prove It

MHF Helper
Aug 2008
12,883
4,999
A horizontal asymptote is NOT a value "f(x) cannot take". If the limit, as x goes to infinity or negative infinity, is a finite number, then y= that number is the horizontal aymptote.
In this case, what you are saying and what I said are one and the same. But you are correct in the general case.