Question

How many inflexion points does this function have?

`f''(x)=(x-1)^3 (x^2 -4)(x^2 +1/2)(x+1)^2`

How many inflexion point does function f have? I believe it has 5 but i need someone to confirm it please.

Answer 1

See below.

Explanation:

If you are basing this on `f''(x)=0`, there are 4 real roots and 2 complex roots to this.

`1 , 2 , -2 , -1 , 1/2 isqrt(2) , -1/2 isqrt(2)`

It should be remembered that when `f''(x)=0` signifies
max/min or points of inflexion, so you can only be sure if you test
each value to the left and right. If there is no change in the sign then it's a point of inflexion.

This is the shape of the graph of the original function. This may or may not need translating up or down, since I can only find this function from an indefinite integral. You can see though that points of inflexion are not immediately obvious.

Answer 2

`f` has three inflection points.

Explanation:

`f''(x)` changes sign at every zero of odd multiplicity.

That is: at `1`, `2`, and `-2`

The factors `x^2+1/2` and `(x+1)^2` are always positive.

So the concavity of `f` depends only on the sign of `(x-1)^3(x-2)(x+2)`

`f` is concave up on `(-2,1)` and on `(2,oo)`

And `f` is concave down on `(-oo,-2)` and on `(1,2)`.