find values of x

Question

Find values of x such that f(x)>0 and all x such that f(x)<0, and sketch the graph of f.

f(x)=x^3-5x^2-4x+20

f(x)>0:

f(x)<0:

Answer 1

f(x) = x^3 -5x^2 -4x + 20 = (x-2)(x+2)(x-5)

roots or x intercepts are -2, 2 and 5

f(x)> 0 when -2<x<2 and when x>5

f(x)<0 when x<-2 and when 2<x<5

the graph of f(x) intersects the x axis at -2, 2 and 5

local maximum is between x= -2 and 2, about 20 or more

local minimum is between x=2 and x=5, maybe -12 or so

after x=5, f(x) increases without limit

less than x=-2, f(x) decreases without limit