0 = 12x^3 + 3(k-18)x^2 + (64-12kx) + 8k-48

The DELTA discriminant for whether this cubic equation has:

3 distinct real roots when DELTA > 0

multiple root with all roots real when DELTA = 0

or one real root and two complex conjugate roots when DELTA < 0

(Source: Wikipedia entry for cubic polynomial equations)

Using an algebraic package you can solve this for k where DELTA = 0

So DELTA = 432k^4 + 12096k^3 + 83520k^2 - 345600k - 3998208

I then used a grapical calculator with a polynomial of order 4 solver to get numerical solutions:

Two of them were complex conjugates of (-12.1161977552, +/- 3.62069124665) these are irrelevant for this analysis.

The real solutions are relevant, one of them was Bob's k = 5.9536152399

The other was also mentioned by Bob earlier to less accuracy and I am getting: k = -9.72121972948

These solutions for k indicate that there is a multiple root with all roots real and a graph shows that they have two stationary points

in terms of the original quartic expression involving x and k. One is an inflection and the other a minimum.

You could work out whether there are 3 turning points or 1 turning point for the ranges either side and in between these

if you wanted to give a full analysis of how many turning points you get for all values of real k.

(Perhaps draw a graph on a graphics calulator for the function of DELTA in terms of k and see where it is above zero

and where it is below zero. Then use the wiki quote that I gave above.)

If I am understanding this correctly there is one stationary point inbetween the two (k>-9.7212197... and k<5.9536152...)

and there are three stationary points for k < -9.721297... and for k > 5.9536152...

The exact formula for calculating the roots of k is extremely complicated and I would not like to attempt that one

without a something like Wolfram or another computer algebra package.

I was referring to the topic where we were having the antonym discussion.

]]>You want me to respond to a topic called 'Discrete Calculus'. This didn't ring any bells in my brain so I did a search. No topic found.

So I looked through the recent topics list. Still nothing.

So I asked for the url and you have said it is the same one. Same as what. Is this some sort of test? Can I see the invisible words? Am I expected to read your mind? Is there a secret code?

dictionary.com wrote:

dis·creet

[dih-skreet] adjective

1.

judicious in one's conduct or speech, especially with regard to respecting privacy or maintaining silence about something of a delicate nature; prudent; circumspect............................

ibid wrote:

cal·cu·lus

[kal-kyuh-luhs] noun, plural cal·cu·li [kal-kyuh-lahy]

1.

.....................2.

Pathology . a stone, or concretion, formed in the gallbladder, kidneys, or other parts of the body.

I get it. You are worried about my kidney stone but are trying to be discreet about it. Too late! I've told everyone now.

Bob

]]>you haven't yet replied to the Discrete Calculus topic

What is this topic?

url?

Bob

]]>By the way, you haven't yet replied to the Discrete Calculus topic and our "interesting" antonym discussion.

Antonym reply done.

Cannot find the other ??? It is being very discreet.

Bob

]]>By the way, you haven't yet replied to the Discrete Calculus topic and our "interesting" antonym discussion.

]]>

You did that on paper of course.

But it's nice to know there's an exact solution.

Thanks.

Bob

]]>I don't think you'll like seeing this one:

]]>Bob

]]>Did you try k= 5.95361523..... ?

Bob

]]>Well I thought I had. See post 4

By all means tell me where I'm going wrong.

Bob

]]>