1. ## Decreasing function

Question:

The diagram shows the curve $y = x^3 - 3x^2 - 9x + k$, where k is constant. The curve has a minimum point on the x-axis.
(i) Find the value of $k$.
(ii) Find the coordinates of the maximum point of the curve.
(iii) State the set of values of $x$ for which $x^3 - 3x^2 - 9x + k$ is a decreasing function of $x$.

Attempt:
(i) $k = 27$
(ii) $(-1,32)$
(iii) I don't know how to get the decreasing order?!...

2. Keep in mind that the function is decreasing before it hits that minimum.

3. Originally Posted by colby2152
Keep in mind that the function is decreasing before it hits that minimum.
Thanks colby2152, but I don't know which formula to use!..

4. Originally Posted by looi76
Question:

The diagram shows the curve $y = x^3 - 3x^2 - 9x + k$, where k is constant. The curve has a minimum point on the x-axis.
(i) Find the value of $k$.
(ii) Find the coordinates of the maximum point of the curve.
(iii) State the set of values of $x$ for which $x^3 - 3x^2 - 9x + k$ is a decreasing function of $x$.

Attempt:
(i) $k = 27$
(ii) $(-1,32)$
(iii) I don't know how to get the decreasing order?!...
Well actually its quite clear from the graph that it is decreasing between the maximum and the minimum. So for -1 < x < 3, we have the function decreasing....

A calculus way of showing it is computing the derivative of f(x) and if the derivative is less than 0 then the function is decreasing.

So $f'(x) = 3x^2 - 6x -9 < 0 \Rightarrow 3(x^2 - 2x - 3) = 3(x-3)(x+1) < 0 \Rightarrow x \in (-1,3)$