math
posted by FUNCTIONS .
Determine an expression for f(x) in which:
f(x) is cubic,
f(x) ¡Ý 0 when x ¡Ü 2,
f(x) < 0 when x > 2, and
f(0) = 4

I assume you meant:
f(x) >= 0 when x <= 2,
f(x) < 0 when x > 2, and
f(0) = 4
We know that there is a root at x=2, so
f(x) = a(x2)(bx^2+cx+d)
f(0) = a(2)(d) = 2ad = 4
That's all we know. So, if we let a=1, d = 2 and
f(x) = (x2)(bx^2+cx2)
Unless you have some other info, b and c can be anything.
If I got the definition of f(x) wrong, fix it and follow the logic.