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

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Answer 1 · 2016-11-30T07:10:57

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

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]}

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