Question:

The diagram shows the curve $\displaystyle y = x^3 - 3x^2 - 9x + k$, where k is constant. The curve has a minimum point on the x-axis.

(i) Find the value of $\displaystyle k$.

(ii) Find the coordinates of the maximum point of the curve.

(iii) State the set of values of $\displaystyle x$ for which $\displaystyle x^3 - 3x^2 - 9x + k$ is a decreasing function of $\displaystyle x$.

Attempt:

(i) $\displaystyle k = 27$

(ii) $\displaystyle (-1,32)$

(iii) I don't know how to get the decreasing order?!...