Question

How do you find the minimum and maximum value of `y=-(x-1)(x+4)`?

Answer 1

`y=-(x-1)(x+4) = -(x^2 + 3x - 4)`

Completing the square:

`= -((x + 3/2)^2 - 9 /4- 4)`

`= -((x + 3/2)^2 - 25 /4 )`

`implies y = -(x + 3/2)^2 + 25 /4`

Now:

`(x + 3/2)^2 ge 0 qquad forall x`

`:. - (x + 3/2)^2 le 0 qquad forall x`

`:. y le 25/4 qquad forall x`

`implies {(y_("max") = 25/4),(y_("max") = - oo):}`

graph{-(x-1)(x+4) [-5, 5, -9.01, 9.01]}