Question

What are the points of inflection of #f(x)=1/(5x^2+3) #?

Answer 1

No inflection points

Explanation:

Calculate the zeroes of `f'(x)`. The points of inflection are these points where `f'(x)=0` and also `f''(x)=0`.

`f'(x)=-(10x)/(5x^2+3)^2`

`f'(x)=0 rArr -10x=0 rArr x=0`, a single possible point of inflection. Note that both `f` and `f'` are well-defined at this point.

Calculate `f''(x)` to determine the nature of the point:

`f''(x)=(-10(5x^2+3)^2+10x(2(5x^2+3)*10x))/(5x^2+3)^4`
`=(-10(5x^2+3)^2+200x^2(5x^2+3))/(5x^2+3)^4`
`=(-10(5x^2+3)+200x^2)/(5x^2+3)^3`

Therefore `f''(0)=(-30+0)/3^3=-30/27<0`. So this point is a local maximum of `f`, not a point of inflection.

Compare with the graph of the function for a visual sanity check:
graph{1/(5x^2+3) [-2.5, 2.5, -1.25, 1.25]}