- #1
olyviab
- 11
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Homework Statement
f(x) = x^3/(x^2+ 1)
1. Identify Domain and Range
2. x and y intercepts (if any)
3. local and global extrema
4. Equations of the asymptotes (if any)
Homework Equations
f(x) = x^3/(x^2+ 1)
The Attempt at a Solution
f(x) = x^3/(x^2+ 1)
f'(x) = (x^2)(x^2 + 3)/(x^2+ 1)^2
1. Identify Domain and Range
2. x and y intercepts (if any)
- I found that the intercept was at x=0 and y=0 ; [0,0]
3. local and global extrema
Critical Points:
0 = x^2 + 1
x = [tex]\pm[/tex]1
0 = x^2
x = 0
0 = x^2 + 3
x = [tex]\sqrt{-3}[/tex]
(critical points) -1 0 1
(points to test) -2 -.5 .5 2
(increasing/decreasing) (+) (+) (+) (+)
I found that there was no extrema by doing the first derivative test.
4. Equations of the asymptotes (if any)
f(x) = x^3/(x^2+ 1)
because the degree > degree of denominator there are no horizontal asymptotes and the function spike to + infinity
derivative indicates the function is always increasing , bottom of the function is always positive , thus the top defines the behavior ---> x is all reals and y is all reals...
dividing yields f(x) = x - { x / [x² + 1 ] }---> y = x is " slant asymptote "
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