Question

If `(7x^2)/(2x-3)`, what are the points of inflection, concavity and critical points?

Answer 1

There is a local maximum at `(0,0)` and a local minimum at `(3,21)`
No inflection point.
The function is concave in the interval `(-oo,3/2)` and convex in the interval `(3/2,+oo)`

Explanation:

We need

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

We calculate the first and second derivatives

Let `f(x)=(7x^2)/(2x-3)`

`u=7x^2`, `=>`, `u'=14x`

`v=2x-3`, `=>`, `v'=2`

Therefore,

`f'(x)=(14x(2x-3)-7x^2(2))/(2x-3)^2`

`=(28x^2-42x-14x^2)/(2x-3)^2`

`=(14x^2-42x)/(2x-3)^2`

`=(14x(x-3))/(2x-3)^2`

The critical points are when `x=0` and `x=3`

We can build a chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `0` `color(white)(aaaaaaaa)` `3/2` `color(white)(aaaaaaa)` `3` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-3` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

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

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

There is a local maximum at `(0,0)` and a local minimum at `(3,21)`

Now, we determine the second derivative

`u=(14x^2-42x)`, `=>`, `u'=28x-42`

`v=(2x-3)^2`, `=>`, `v'=4(2x-3)`

`f''(x)=((28x-42)(2x-3)^2-4(14x^2-42x)(2x-3))/(2x-3)^4`

`=((2x-3)(56x^2-84x-84x+126-56x^2+168x))/(2x-3)^4`

`=126/(2x-3)^3`

There is no inflexion point.

We can build a chart

`color(white)(aaaa)` `Interval` `color(white)(aaaa)` `(-oo,3/2)` `color(white)(aaaa)` `(3/2,+oo)`

`color(white)(aaaa)` `f''(x)` `color(white)(aaaaaaaaaa)` `-` `color(white)(aaaaaaaaaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaaaaaa)` `nn` `color(white)(aaaaaaaaaaa)` `uu`

The function is concave in the interval `(-oo,3/2)` and convex in the interval `(3/2,+oo)`

graph{7x^2/(2x-3) [-66.14, 65.5, -22.55, 43.3]}