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

Jan 16, 2018

At $x = 1$

#### Explanation:

The point of inflection is the point where the second derivative of the function $f \left(x\right)$ is zero i.e. $\frac{{d}^{2} f}{{\mathrm{dx}}^{2}} = 0$. At these points the slope of the curve changes from increasing to decreasing and vice versa.

Here function is $f \left(x\right) = {x}^{3} - 3 {x}^{2} - x + 7$

and $\frac{\mathrm{df}}{\mathrm{dx}} = 3 {x}^{2} - 6 x - 1$ and $\frac{{d}^{2} f}{{\mathrm{dx}}^{2}} = 6 x - 6$

and $\frac{{d}^{2} f}{{\mathrm{dx}}^{2}}$ is zero, when $6 x - 6 - 0$ 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]}