Identifying inflection points and critical points
I'm given an equation and asked to find the following:
- Domain
- Critical points
- Inflection points
- Asymptotes
The function is:
$y = 2 + 9x + 3x^2 -x^3$
It has been quite awhile since I have done this sort of problem so I'm a bit rusty. Can someone check my work and explain the missing parts?
- Domain
All Real
- Critical Points:
Find derivative:
$ y' = -3x^2 + 6x + 9$
Set equal to 0:
$ 3x^2 - 6x - 9 = 0 $
$ (3x + 3)(x - 3) = 0$
$ x = -1, x = 3$
Corresponding points: (-1, -3) & (3,30)
- Inflection Points:
Take second derivative:
$ y'' = 6x - 6$
Set it to 0:
edit (fix typo)
$ x = 1 $
Corresponding point: (1, 13)
- Asymptotes
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1 Answer
$\begingroup$Yes, you find inflection points by taking the second derivative $y''$ and setting $y''$ equal to zero. Solve for x, to determine the point $(x, y)$ at which an inflection point may occur. (This procedure may not result in an inflection point, but in this case it does. If an inflection point exists, it will be at the point at which $y'' = 0$, but the converse is not always true.
For your critical points, find the corresponding $y$ values associated with your solutions fr $x$, respectively, to obtain the critical points: you want to write the points as ordered pairs.
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