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.

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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

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