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 January 17th, 2014, 12:49 PM #1 Banned Camp   Joined: Dec 2013 Posts: 34 Thanks: 1 How many critical numbers in this function? $f'(x) = 1 + \frac{210\sin x}{x^{2}-6x+10}$ From my textbook: "A critical number of a function f is a number in c in the domain of f such that either f'(c)=0 or f'(c) does not exist." This function is the derivative of the function whose **quantity of** critical numbers I am supposed to find (the explanation of the problem set confirms this as well). Technically, I am supposed to use a Computer Algebra System to count how many exist, not where they are exactly, but even that has become difficult. How can I find them?
 January 17th, 2014, 12:54 PM #2 Banned Camp   Joined: Dec 2013 Posts: 34 Thanks: 1 Re: How many critical numbers in this function? btw, the answer is 16--there are 16 critical numbers to the given function's...antiderivative? This section precedes integration/antiderivatives, btw. My biggest problem is that when I graph f'(x) (the given function), it does not appear to intersect the x-axis enough times. What's going on?
 January 17th, 2014, 01:22 PM #3 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,976 Thanks: 1158 Math Focus: Elementary mathematics and beyond Re: How many critical numbers in this function? f '(x) = (x^2 - 6x + 10 + 210sinx)/(x^2 - 6x + 10) = 0. Assuming you have a graphing calculator, graph -210sinx and x^2 - 6x + 10 in the same window with Xmin = -20, Xmax = 20, Xscl = 2, Ymin = -230, Ymax = 230, Yscl = 23. I count 10 points of intersection. Since the quadratic has no real roots, there are no undefined values of the derivative. What is the function that you have given the derivative of?
 January 17th, 2014, 01:38 PM #4 Banned Camp   Joined: Dec 2013 Posts: 34 Thanks: 1 Re: How many critical numbers in this function? no, see, I did a Wolfram graph, too (lol), and I already know the only two roots of the denominator are imaginary, so the function is continuous. Only 10 intersections are shown. Only the derivative was provided in this problem. It is from the 6th edition of James Stewart's Calculus with Selected Classics Problem Sets (Calculus Classics Edition). It's an all-encompassing compilation--a really big book, 3 inches thick, possibly 3 volumes-worth of material (because when I bought it, it was used for all 3 semesters of Calculus), though it's not divided into volumes. Where are the other 6 critical points coming from?
 January 17th, 2014, 09:03 PM #5 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234 Re: How many critical numbers in this function? Are you sure it's not 710sinx ? http://www.wolframalpha.com/input/?i=71 ... 2%2B6+x-10
January 22nd, 2014, 12:20 AM   #6
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Re: How many critical numbers in this function?

Quote:
 Originally Posted by agentredlum Are you sure it's not 710sinx ? http://www.wolframalpha.com/input/?i=71 ... 2%2B6+x-10
Yes, I'm sure. I like your thinking, "wouldn't it be nice if the sine function was scaled more to intersect the polynomial few more times?"
But no, the coefficient is 210, not 710. Have you seen this same question in another edition with 710?

 January 22nd, 2014, 10:50 AM #7 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234 Re: How many critical numbers in this function? I don't remember seeing this anywhere and it's been years since i held a James Stewart book. IMHO he is a wonderful author. I'm glad you appreciate my attempt , it seemed the easiest way to get it to work once i suspected there may be a misprint , just change one number from 2 to 7. I played around with it for a while till the idea came to me.

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