Show the work needed to determine the x-coordintes for all points of inflection for some continuous function g(x) if we are given that... Show the work needed to determine the x-coordintes...
Show the work needed to determine the x-coordintes for all points of inflection for some continuous function g(x) if we are given that...
Show the work needed to determine the x-coordintes for all points of inflection for some continuous function g(x) if we are given that...
g'(x) = (3x^2+15x+6)/(x-1)^(1/3) and we are told that the domain of g(x) is all real numbers.
- print Print
- list Cite
Expert Answers
calendarEducator since 2011
write3,002 answers
starTop subjects are Math, Science, and Business
We want to determine the points of inflection for a function given that its derivative is `g'(x)=(3x^2+15x+6)/((x-1)^(1/3))` and that the function's domain is `RR` .
To find the points of inflection we find the second derivative and note where the concavity of the graph changes -- i.e. when the sign of the second derivative changes.
`g''(x)=((x-1)^(1/3)(6x+15)-(3x^2+15x+6)(1/3)(x-1)^(-2/3))/((x-1)^(2/3))`
`=((x-1)^(-2/3)[(x-1)(6x+15)-(3x^2+15x+6)(1/3)])/((x-1)^(2/3))`
`=(5x^2+4x-17)/((x-1)^(4/3))`
Now `g''(x)` fails to exist at x=1, and is zero at `x=(-4+-sqrt(356))/10` or `x=-2/5+-sqrt(89)/5 ~~-2.287,1.487`
So we check the sign of the second derivative on the following intervals:
`(-oo,-2/5-sqrt(89)/5):` the second derivative is positive so the function is concave up on this interval.
`(-2/5-sqrt(89))/5,1):` the second derivative is negative so the function is concave down on this interval.
`(1,-2/5+sqrt(89)/5):` the second derivative is negative so the function is concave down on this interval.
`(-2/5+sqrt(89)/5,oo):` the second derivative is positive so the function is concave up on this interval.
The sign of the second derivative changes at `x=-2/5-sqrt(89)/5` and at `x=-2/5+sqrt(89)/5` so these are the inflection points.
The graph of a possible g(x) (the function could be shifted vertically without changing points of inflection, extrema, etc...):
Note that the first derivative fails at x=1 as there is a cusp there. Also, the shift from concave down to concave up at `x~~1.5` is very difficult to see from the graph.
Related Questions
- Given f(x) and g(x), please find (fog)(X) and (gof)(x) f(x) = 2x g(x) = x+3
- 1 Educator Answer
- Given f (x) = 6x^2 + x and g(x) = 3− 2x , find all values of x such that f (x) = g(x)?
- 2 Educator Answers
- If f and g are continuous functions prove that M(x)=max{f(x),g(x)} and m(x)=min{f(x),g(x)} are...
- 1 Educator Answer
- For the given functions f , g, and h, find `fogoh` and state the exact domain of `fogoh` ....
- 1 Educator Answer
- Given the function f(x)=2x-3, find the function g(x)=f(x+1)+f(x-1)?
- 1 Educator Answer
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.