Question

For what values of x is `f(x)=4x^5-5x^4` concave or convex?

Answer 1

The answer is `f(x)` is concave down for `x in ]-oo, 1]` and concave up when `x in [1, +oo[`

Explanation:

We calculate the first derivative and we build a sign chart

`f(x)=4x^5-5x^4`

`f'(x)=20x^4-20x^3`

`f'(x)=20x^3(x-1)`

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

`20x^3(x-1)=0`

`x=0` and `x=1`

Now we construct the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaaa)` `0` `color(white)(aaaaaa)` `1` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaa)` `-` `color(white)(aa)` ###color(white)(aaaa)#`+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-1` `color(white)(aaaaa)` `-` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+`

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

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `↗^(0)` `color(white)(aaaa)` `↘_(-1)` `color(white)(aaaa)` `↗^(+oo)`

Therefore,

`f(x)` is concave down for `x in ]-oo, 1]` and concave up when

`x in [1, +oo[`