Question

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

Answer 1

`f(x)` has one point of inflection at `x approx .3288`

Explanation:

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

`f'(x)=3x^2-14x+ color(blue)(2xe^x)+2e^x`

`f''(x)=6x-14+ color(blue)(2xe^x+2e^x)+2e^x`

`f''(x)=6x-14+ 2xe^x+4e^x`

To find points of inflection, set `f''(x)` equal to zero.
`0=2[3x-7+xe^x+4e^x]`

(using a graphing calculator):
`x approx .3288`