Do you need to show algebraic steps? I think all can be explained by describing the graph.From consideration of the sketch of y = f(x) show that if the equation f(x) = k has three real roots then k must be negative. Give the sign of each root in this case.
Consider this: if you took the graph and do a vertical translation upward, then the graph would have 3 x-intercepts, or 3 real roots. But you can't shift the graph up too much, because you have a local minimum at (5/3, -64/27). Basically, given that
as long as 0 < c < 64/27,
will have 3 real roots. To solve such an equation, you would set it equal to 0:
Comparing this to , k = -c, and since c is in the interval (0, 64/27), k is negative.
If we are limiting our c to 0 < c < 64/27, then the 3 real roots are positive.
Also, see diagram. The black curve is y = f(x), the green curve is y = f(x) + 1, and the blue curve is y = f(x) + 64/27. The green curve has 3 real roots. The last curve doesn't give us 3 real roots because now the local minimum touches the x-axis. That's why when doing a vertical translation f(x) + c, c has to be less than 64/27.