Question

For what values of x is `f(x)=(2x-1)(3x-5) (x-2)` concave or convex?

Answer 1

`f(x)` is concave when `x in ]-oo,25/18[` and convex when `x in ]25/18,+oo[`

Explanation:

We develop the expression and we calculate the first and second derivatives

`f(x)=(2x-1)(3x-5)(x-2)`

`=(6x^2-13x+5)(x-2)`

`=6x^3-12x^2-13x^2+26x+5x-10`

`f(x)=6x^3-25x^2+31x-10`

`f'(x)=18x^2-50x+31`

`f''(x)=36x-50`

`f''(x)=0` when `x=50/36=25/18`

We construct a table

`color(white)(aaaa)` `Interval` `color(white)(aaaa)` `|` `color(white)(aaaa)` `]-oo,25/18[` `color(white)(aaaa)` `|` `color(white)(aaa)` `]25/18,+oo[` `|`

`color(white)(aaaa)` `Sign f''(x)` `color(white)(aa)` `|` `color(white)(aaaaaaaa)` `-` `color(white)(aaaaaaa)` `|` `color(white)(aaaaaaaa)` `+` `color(white)(aa)` `|`

`color(white)(aaaa)` `function` `color(white)(aaaa)` `|` `color(white)(aaaaaaaaa)` `nnn` `color(white)(aaaaaa)` `|` `color(white)(aaaaaaaa)` `uuu` `color(white)(aa)` `|`

Therefore,

`f(x)` is concave when `x in ]-oo,25/18[` and convex when `x in ]25/18,+oo[`