Question

How do you find the inflection points for `f(x)=x^4-10x^3+24x^2+3x+5`?

Answer 1

Inflection points are points of the graph of `f` at which the concavity changes.
In order to investigate concavity, we look at the sign of the second derivative:

`f(x)=x^4-10x^3+24x^2+3x+5`

`f'(x)= 4x^3-30x^2+48x+3`

`f(x)=12x^2-60x+48 = 12(x^2-5x+4) = 12(x-1)(x-4)`

So, `f''` never fails to exist, and `f''(x)=0` at `x=1, 4`

Consider the intervals:

`(-oo,1)`, `f''(x)` is positive, so `f` is concave up
`(1,4)`, `f''(x)` is negative, so `f` is concave down
`(4,oo)`, `f''(x)` is positive, so `f` is concave up

The concavity changes at `x=1` and `y = f(1) = 23`.

The concavity changed again at `x=4` and `y = f(4) = 17`.

The inflection points are:

`(1,23)` and `(4, 17)`.