A) The function is expressed in the factored form. In factored form the quadratic function written as a product of two terms as follows:
f(x)= (x-x1)(x-x2) such that x1 and x2 are the roots.
In n(12-n) = (n-0)(12-n) ==> Then the roots are 0 and 12.
`B)==> n(12-n)= 0 `
`==> n= 0 and n= 12`
C) To find the vertex, we will rewrite the function in the standard form `ax^2 + bx + c.`
`p(n)= n(12-n)= -n^2 +12n`
`==> a = -1, b= 12, and c = 0`
`D). V(v_x, v_y)`
`v_x = -b/(2a) = -12/-2 = 6`
`v_y= -(b^2-4ac)/4a = -(144)/-4 = 36`
`==> V(-6, 36)`
The factor of n^2 is negative ( a<0 ), then the function has a maximum point at the vertex.