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?
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............................
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
]]>