Question

For what values of x is `f(x)=(2x^2−4x)e^x` concave or convex?

Answer 1

The function is convex for `x in (-oo,-1-sqrt3) uu(-1+sqrt3, +oo)` and concave for `x in (-1-sqrt3, -1+sqrt3)`

Explanation:

`"Reminder"`

`(uv)'=u'v+uv'`

Calculate the first and second derivatives

`f(x)=(2x^2-4x)e^x=2(x^2-2x)e^x`

`f'(x)=2*(2x-2)e^x+2(x^2-2x)e^x=2(x^2-2)e^x`

`f''(x)=2*(2x)e^x+2(x^2-2)e^x=2(x^2+2x-2)e^x`

The inflection points are when `f''(x)=0`

`x^2+2x-2=0`

Solving this quadratic equation for `x`

`x=(-2+-sqrt(2^2-4*1*(-2)))/(2)=(-2+-sqrt(12))/(2)`

`=(-2+-2sqrt(3))/2`

`=-1+-sqrt3`

The roots are

`x_1=-1-sqrt3`

`x_2=-1+sqrt3`

We can build the variation chart

`color(white)(aaa)` `color(white)(aaa)` `"Interval"` `color(white)(aaa)` `(-oo, x_1)` `color(white)(aaa)` `(x_1, x_2)` `color(white)(aaa)` `(x_2, +oo)`

`color(white)(aaa)` `color(white)(aaa)` `"Sign f''(x)"` `color(white)(aaaaa)` `+` `color(white)(aaaaaaaa)` `-` `color(white)(aaaaaaa)` `+`

`color(white)(aaa)` `color(white)(aaa)` `" f(x)"` `color(white)(aaaaaaaaa)` `uu` `color(white)(aaaaaaaa)` `nn` `color(white)(aaaaaaa)` `uu`

graph{(2x^2-4x)e^x [-17.35, 14.68, -7.95, 8.07]}