how to find inflection point

`Inflection Point`

An inflection point is a point where the graph of the function changes CONCAVITY (from up to down or vice versa).

It could be seen as a Switching point, which means the point that the Slope of function switch from increasing and decreasing.
e.g., the function might be still going up, but at such a point it suddenly increases slower and slower. And we call that point an inflection point.

Refer to Khan academy video for more intuition rapidly: Inflection points from graphs of function & derivatives

Algebraically, we identify and express this point by the function's First DerivativeOR Second Derivative.

Example

Intuitive way to solve:

Draw a tangent line in imagination and move it on the function from left to right
Notice the tangent line's slope, does it go faster or slower or suddenly change its pace at a point?
We found it suddenly changed at point c.

More definitional way to solve:

Looking for the parts of concavity shapes
Seems that B-C is a part of Concave Down, and C-D is a part of Concave Up
So C is a SWITCHING POINT, it's a inflection point.

Example

Solve:

Looking for the parts of concavity shapes
There's no changing of concavity shapes, there's only one shape: Concave down.

Example

Solve:

Notice that's the graph of f'(x) , which is the First Derivative.
Checking Inflection point from 1st Derivative is easy: just to look at the change of direction.
Obviously there're only two points changed direction: -1 & 2

Example

Solve:

Mind that this is the graph of f''(x) , which is the Second derivative.
Checking inflection points from 2nd derivative is even easier: just to look at when it changes its sign, or say crosses the X-axis.
Obviously, it crosses the X-axis 5 times. So there're 5 inflection points of f(x).

Example: Finding Inflection points

Solve:

Function has POSSIBLE inflection points when f''(x) = 0.
Set f''(x) =0 and solve for x, got x=-3.
We now know the possible point, but don't know its CONCAVITY. This need to try some numbers from its both sides:

So it didn't change the concavity at point -3, means there's no inflection point for function.

Example: Finding Inflection points

Solve:

Function has POSSIBLE inflection points when f''(x) = 0.
Set f''(x) =0 and solve for x, got x=0 or 6.

Refer to Symbolab for f''(x).)
Refer to Symbolab for f''(x)=0.

We now know the possible point, but don't know its CONCAVITY. This need to try some numbers from its both sides:

So it didn't change the concavity at point 0, means only 6 is the inflection point.