Question

What are the points of inflection of `f(x)=x^5 - 2x^3 +4x`?

Answer 1

The points of inflection of `f(x)` are at
`x=-sqrt(3/10), x=0,` and `x=sqrt(3/10)`

Explanation:

Use power rule to find `f''(x)`

`f(x)=x^5-2x^3+4x`
`f'(x)=5x^4-6x^2+4`
`f''(x)=20x^3-6x`

For points of inflection, set second derivative equal to zero:
`0=f''(x)`
`0=2(x)(10x^2-3)`

Using zero product rule :
`color(blue)(x=0`

`10x^2-3=0`
`x^2=3/10`
`color(blue)(x=+-sqrt(3/10)`

Double check that these answers are actual points of inflection by testing values in between these intervals (or draw a "sign line", as some call it).

Thus, the points of inflection of `f(x)` are at
`x=-sqrt(3/10), x=0,` and `x=sqrt(3/10)`