Question

If `y= (x^2+1)/(x^2-4)`, what are the points of inflection, concavity and critical points?

Answer 1

The critical point is at `(0, -1/4)` and it is a local maximum.
The intervals of convexity are `(-oo, -2) uu (2,+oo)`
The interval of concavity is `(-2,2)`

Explanation:

Calculating the first and second derivatives and building the variation charts

`y=(x^2+1)/(x^2-4)`

The domain of `y` is `x in {RR -(-2,2)}`

`u(x)=x^2+1`, `=>`, `u'(x)=2x`

`v(x)=x^2-4`, `=>`, `v'(x)=2x`

Therefore,

`dy/dx=(u'v-uv')/(v^2)=(2x(x^2-4)-2x(x^2+1))/(x^2-4)^2`

`=(2x^3-8x-2x^3-2x)/(x^2-4)^2`

`=(-10x)/(x^2-4)^2`

The critical points are when `dy/dx=0`

`=>`, `-10x=0`, `x=0`

Building the variation chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `-2` `color(white)(aaaaaaaa)` `0` `color(white)(aaaaaaaaa)` `2` `color(white)(aaaaaaa)` `+oo`

`color(white)(aaaa)` `dy/dx` `color(white)(aaaaaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaa)` `+` `color(white)(aaa)` `0` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `-`

`color(white)(aaaa)` `y` `color(white)(aaaaaaaaa)` `↗` `color(white)(aaa)` `||` `color(white)(aaa)` `↗` `color(white)(a)` `-1/4` `color(white)(aaaa)` `↘` `color(white)(aaa)` `||` `color(white)(aaaa)` `↘`

Calculating the second derivative to determine the concavity and the points of inflections

`u(x)=-10x`, `=>`, `u'(x)=-10`

`v(x)=(x^2-4)^2`, `=>`, `v'(x)=4x(x^2-4)`

`(d^2y)/(dx^2)=(u'v-uv')/(v^2)=(-10(x^2-4)^2+40x^2(x^2-4))/(x^2-4)^4`

`=((x^2-4)(-10x^2+40+40x^2))/(x^2-4)^4`

`=(10(3x^2+4))/(x^2-4)^3`

When `x=0`, `=>`, `(d^2y)/(dx^2)<0`, this is a local max

Building the variation chart

`color(white)(aaaa)` `Interval` `color(white)(aaaa)` `(-oo,-2)` `color(white)(aaaa)` `(-2,2)` `color(white)(aaaa)` `(2,+oo)`

`color(white)(aaaa)` `Sign (d^2y)/(dx^2)` `color(white)(aaaaaaaa)` `+` `color(white)(aaaaaaaaaa)` `-` `color(white)(aaaaaaaa)` `+`

`color(white)(aaaa)` `y` `color(white)(aaaaaaaaaaaaaaa)` `uu` `color(white)(aaaaaaaaaa)` `nn` `color(white)(aaaaaaaa)` `uu`

See the graph below

graph{(x^2+1)/(x^2-4) [-7.9, 7.9, -3.95, 3.95]}