Can a point of inflection be undefined?

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Answer 1 · 2015-10-21T14:03:34

See the explanation section below.

A point of inflection is a point on the graph at which the concavity of the graph changes.

If a function is undefined at some value of $x$ $x$ , there can be no inflection point.

However, concavity can change as we pass, left to right across an $x$ $x$ values for which the function is undefined.

Example

$f (x) = \frac{1}{x}$ $f(x) = 1/x$ is concave down for $x < 0$ $x < 0$ and concave up for $x > 0$ $x > 0$ .

The concavity changes "at" $x = 0$ $x=0$ .

But, since $f (0)$ $f(0)$ is undefined, there is no inflection point for the graph of this function.

graph{1/x [-10.6, 11.9, -5.985, 5.265]}

Example 2

$f (x) = \sqrt[3]{x}$ $f(x) = root3x$ is concave up for $x < 0$ $x < 0$ and concave down for $x > 0$ $x > 0$ .

$f'(x) =1/(3root3x^2)$ and $f'(x) =(-2)/(9root3x^5)$

The second derivative is undefined at $x=0$ .

But, since $f(0)$ is defined, there is an inflection point for the graph of this function. Namely, $(0,0)$

graph{x^(1/3) [-3.735, 5.034, -2.55, 1.835]}