Question

What are the points of inflection, if any, of `f(x) =-3x^3 - 7x^2 + 3x`?

Answer 1

The point of inflection is `=(-0.778,-5.156)`#

Explanation:

We calculate the first and second derivatives

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

`f'(x)=-9x^2-14x+3`

`f''(x)=-18x-14`

The point of inflection is when

`f''(x)=0`

`-18x-14=0`, `=>`, `x=-14/18=-7/9`

Therefore, the point of inflection is

`(-7/9, f(-7/9))=(-0.778,-5.156)`

We can build a chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `(-oo, -7/9)` `color(white)(aaaa)` `(-7/9,+oo)`

`color(white)(aaaa)` `f''(x)` `color(white)(aaaaaa)` `+` `color(white)(aaaaaaaaaaaaa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaa)` `uu` `color(white)(aaaaaaaaaaaaa)` `nn`
graph{(y-(-3x^3-7x^2+3x))((x+0.778)^2+(y+5.156)^2-0.01)=0 [-11.24, 6.54, -8.084, 0.805]}