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

Nov 30, 2016

No point of inflection.
Concavity upwards.
Critical point at $\left(- 1 , 2\right)$

#### Explanation:

Let's calculate the derivatives

$y = {x}^{2} + 2 x + 3$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 x + 2$

Critical points, when $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

$2 x + 2 = 0$

$x = - 1$

Second derivative

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = 2$

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 > 0$, so the concavity is upwards

As $\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 \ne 0$ there are no points of inflection.

Let's do a sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 1$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$\frac{\mathrm{dy}}{\mathrm{dx}}$$\textcolor{w h i t e}{a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$y$$\textcolor{w h i t e}{a a a a a a a a a}$$\downarrow$$\textcolor{w h i t e}{a a a a}$$\uparrow$

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