Points of Inflection

A point of inflection is a point on the graph of a function where the concavity of the graph changes. In other words, it is where a curve transitions from being concave up (shaped like $\cup$ ) to concave down (shaped like $\cap$ ), or vice versa.

A function is concave up when its graph bends upwards, and the second derivative is positive $f''(x)>0$ .
A function is concave down when its graph bends downward, and the second derivative is negative $f''(x)<0$ .
At the point of inflection, the concavity changes which implies that $f''(x)$ either crosses zero or becomes undefined.

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Finding the point of inflection

Given a function $f(x)$ , compute the second derivative $f''(x)$ .
Find where $f''(x)=0$ or $f''(x)$ is undefined.

Example

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Find the points of inflection of the curve $y=x^3-6x^2+5x-11$ .

\frac{dy}{dx}=3x^2-12x+5

\frac{d^2y}{d^2x}=6x-12

Find the point for which $\frac{d^2y}{d^2x}=0$ .

6x-12=0

6x=12

x=2

Find the corresponding $y$ coordinate.

y=(2)^3-6(2)^2+5(2)-11=-17

The point of inflection is :

(2,-17)

Example

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Determine if the function $g(x)=\frac{x}{x+1}, x \ne 1$ has a point of inflection.

g'(x)=\frac{1}{(x+1)^2}

g''(x)=-\frac{2}{(x+1)^3}

Find the point for which $g''(x)=0$ .

-\frac{2}{(x+1)^3}=0

-2=0

We've arrived at a contradiction, $-2$ does not equal to $0$ , hence the function $g(x)$ does not have a point of inflection.

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