
equation
Y=X³+12X²+36X is used to find solutions of the equation X³+12X²+36X=k for various values
The maximum point is (6,0)
The minimum point is (2,32)
1) Verify the point (8,32) is on the graph
2) Find the number of real roots of the equation when k is:
i) 0
ii) 20
iii) 20
iv) 40
3) For what value of k does the equation have three real roots? What can you say about these roots?
any help would be fab

I do not understand your question. Using your given polynomial, the point (8, 32) is not a valid coordinate of the graph.

(−8,−32) is on the graph $\displaystyle y = x^3+12x^2+36x$.
So, the yvalue of the local maximum is 0 and the yvalue of the local minimum is is −32. What this means is that for −32 ≤ k ≤ 0, the equation $\displaystyle x^3+12x^2+36x=k$ has three real roots. (In the case k = −32 or 0, two of the real roots are repeated.) For k < −32 or k > 0, the equation $\displaystyle x^3+12x^2+36x=k$ has only one real root.