Decreasing function

**looi76** · May 21st 2008, 06:59 AM

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?!...

**colby2152** · May 21st 2008, 07:10 AM

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

**looi76** · May 21st 2008, 07:12 AM

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!..

**Isomorphism** · May 21st 2008, 08:23 AM

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

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