Question

If `y= x^2 + 2x + 3`, what are the points of inflection, concavity and critical points?

Answer 1

No point of inflection.
Concavity upwards.
Critical point at `(-1,2)`

Explanation:

Let's calculate the derivatives

`y=x^2+2x+3`

`dy/dx=2x+2`

Critical points, when `dy/dx=0`

`2x+2=0`

`x=-1`

Second derivative

`(d^2y)/dx^2=2`

`(d^2y)/dx^2>0`, so the concavity is upwards

As `(d^2y)/dx^2!=0` there are no points of inflection.

Let's do a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-1` `color(white)(aaaa)` `+oo`

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

`color(white)(aaaa)` `y` `color(white)(aaaaaaaaa)` `darr` `color(white)(aaaa)` `uarr`

graph{x^2+2x+3 [-8.49, 7.31, -0.03, 7.87]}