Question

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

Answer 1

The critical point is `=(0,0)`. The inflection point is `=(-1,1/9)`. See below for the concavities.

Explanation:

We need

`(u/v)'=(u'v-uv')/(v^2)`

Let's calculate the first derivative

`f(x)=x^2/(x-2)^2`

`u=x^2`, `=>`, `u'=2x`

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

`f'(x)=(2x*(x-2)^2-x^2*(2(x-2)))/(x-2)^4`

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

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

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

`f'(x)=0`, when `x=0`

We build our first chart

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

`color(white)(aaaa)` `-4x` `color(white)(aaaaaa)` `+` `color(white)(aaaa)` `0` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaa)` `-`

`color(white)(aaaa)` `(x-2)^3` `color(white)(aaa)` `-` `color(white)(aaaa)` `0` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaa)` `+`

`color(white)(aaaa)` `f'(x)` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `0` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `↘` `color(white)(aaa)` `0` `color(white)(aaaa)` `↗` `color(white)(aaa)` `||` `color(white)(aaa)` `↘`

The critical point is when `x=0`

The intervals of increasing are `x in (0,2)` and decreasing are `x in (-oo,0) uu (2,+oo)`

Now, we calculate the second derivative

`u=-4x`, `=>`, `u'=-4`

`v=(x-2)^3`, `=>`, `v'=3(x-2)^2`

So,

`f''(x)=(-4(x-2)^3+4x*3(x-2)^2)/(x-2)^6`

`=(-4(x-2)^2((x-2)-3x))/(x-2)^6`

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

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

`=(8(x+1))/(x-2)^4`

Therefore,

`f''(x) =0` when `x=-1`, this is the inflexion point

The chart is as follows

`color(white)(aaaa)` `I` `color(white)(aaaaaa)` `(-oo,-1)` `color(white)(aaaa)` `(-1,2)` `color(white)(aaaa)` `(2,+oo)`

`color(white)(aaaa)` `f''(x)` `color(white)(aaaaaaa)` `-` `color(white)(aaaaaaaaa)` `+` `color(white)(aaaaaaaa)` `+`

`color(white)(aaaa)` `f''(x)` `color(white)(aaaaaaa)` `nn` `color(white)(aaaaaaaaa)` `uu` `color(white)(aaaaaaaa)` `uu`

The curve is concave is up when `x in (-1,2) uu (2,+oo)`

The curve is concave down when `x in (-oo,-1)`

Also,

`f''(0)>0`, this is a minimum

graph{x^2/(x-2)^2 [-11.36, 11.14, -1.58, 9.67]}