Inflection Point Calculator

What is an Inflection Point?
Inflections Points are Points of Largest Slope or Smallest Slope on a Graph.

How do you find Inflection Points?
Inflection Point are found by finding sign changes of the 2. derivative.

What is an Inflection Point on a Graph?
At Inflections Points a graph changes its concavity. From concave up to concave or vice versa as shown in image below.

The above image shows an Inflection Point. It occurs when concavity changes. It is the Point of Steepest Slope.

How do I find Inflection Points?

Inflections Points are found by finding the zeros of the 2. derivative AND ensuring a change a sign at those zeros.

Example (simple): If $f(x)=x^3+5$ then $f'(x)=3x^2$ and $f''(x)=6x = 0$ when x=0. Since $$f''(x)=6x$$ changes sign at x=0 then f(x) has an Inflection Point x=0. That was not complicated ;-)

Example for no Inflection Point: If $f(x)=x^2+5$ then $f'(x)=2x$ and $f''(x)=2$ Since $$f''(x)=2$$ never changes sign f(x) has no Inflection Point.
That was not complicated either ;-)

Example(advanced): If $f(x)=x^4-6x^3$ then $f'(x)=4x^3-18x^2$ and $f''(x)=12x^2-36x = 12x*(x-3)$ Since $$f''(x)$$ changes sign at both $$x=0,3$$ f(x) has Inflection Points at $$x=0,3$$.
Now try the above Infection Point Calculator a few times.

Online Calculators with Steps (Free)
﻿