Question

How would you find the inflection point and the concavity of `g(x) = (5x - 2.6) / (5x - 6.76)^2`? I know I have to take the 2nd derivative but i'm not sure how because of the odd way this function is set up.?

Answer 1

Use the quotient rule to find `g'(x)`, then simplify if possible, then use the quotient rule again to find `g''(x)`.

`g(x)= (5x-2.6)/(5x-6.76)^2`

`g'(x)=( 5(5x-6.76)^2 - (5x-2.6)2(5x-6.76)(5))/(5x-6.76)^4`

`= ( 5(5x-6.76)[(5x-6.76) - 2(5x-2.6)])/(5x-6.76)^4`

`= ( 5[5x-6.76 - 10x + 5.2])/(5x-6.76)^3`

`g'(x) = ( -5(5x + 1.56))/(5x-6.76)^3`

Differentiate again to get `g''(x)`.

After finding `g''(x)`, proceed as in any other question about concavity.