Determine whether f(x)=(1)/(2)x^(2)-x-9 has a maximum or a minimum value and find that value.

Question

Determine whether
`f(x)=(1)/(2)x^(2)-x-9` has a maximum or a minimum value and find that value.

Answer 1

First of all, recall that every parabola can be written as

`y=ax^2+bx+c,\quad a\\eq 0, b \in \mathbb{R}, c \in \mathbb{R}`

If
`a>0`, the parabola is concave up (and thus it has a minimum value).

If
`a<0`, the parabola is concave down (and thus it has a maximum value).

So, in your case, the parabola in concave up.

The x-coordinate of the minimum can be found using

`x=-(b)/(2a)=-(-1)/(1)=1`

And the y-coordinate will be

`f(1)=(1)/(2)(1)^2-1-9 = -(19)/(2)`

So, the minimum value is -19/2 at x=1.