Question

What are the points of inflection of #f(x)=x^3-3x^2-x+7 #?

Answer 1

At `x=1`

Explanation:

The point of inflection is the point where the second derivative of the function `f(x)` is zero i.e. `(d^2f)/(dx^2)=0`. At these points the slope of the curve changes from increasing to decreasing and vice versa.

Here function is `f(x)=x^3-3x^2-x+7`

and `(df)/(dx)=3x^2-6x-1` and `(d^2f)/(dx^2)=6x-6`

and `(d^2f)/(dx^2)` is zero, when `6x-6-0` or `x=1`

and at this `y=4`

graph{(x^3-3x^2-x+7-y)((x-1)^2+(y-4)^2-0.01)=0 [-5.043, 4.957, 1.54, 6.54]}