Question

Find the inflection points

y=x^2/x^+3


So far I've found y'=6x/(x^2+3)^2 but I'm stuck in finding y''. So far, I'm at 6(x^2+3)^2-24(x^2+3)/(x^2+3)^4 and the next step says y''=6(-3x^2+3)/(x^2+3)^3=0 and I don't understand how.

Thanks

Answer 1

looking at your answer for y' I will assume you meant:
y = x^2/(x^2 + 3)

your first derivative is correct, I suspect you used the quotient rule.
So why not use it also to get y''

<b>y'' = ( (x^2 + 3)^2 (6) - 6x(2)(x^2 + 3) (2x) )/(x^2 + 3)^4</b>
= ( 6(x^2+3)^2 - 24x^2 (x^2+3) )/(x^2 + 3)^4
= ( 6(x^2+3) (x^2 + 3 - 4x^2)/(x^2+3)^4
= 6(3 - 3x^2)/(x^2+3)^4
= 6(3-3x^2)/(x^2+3)^3

Your error was in the line I printed in bold.

Answer 2

assuming you mean
y = x^2/(x^2+3)
then
y' = 6x/(x^2+3)^2 is correct

y" = [6(x^2+3)^2 - 6x 2(x^2+3)2x ]/(x^2+3)^4

= [6(x^2+3) -24 x^2 ]/(x^2+3)^3
there is your error 6*x*2*2*x = 24 x^2 not 24

= 6 x^2 + 18 -24 x^2 is numerator

-18 x^2 + 18 = 0 for inflection point

x = +/-1