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# 1.What is the upper bound of the function?

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1.What is the upper bound of the function?

f(x)=4x^4−25x^2−5x−13

upper bound =

wouldnt than answer be just 13 since when you do long divsion  13/4 -25 -5 -13 it is 4 27 346 and 4485 which are all postitive ?

Sep 12, 2019
edited by jjennylove  Sep 12, 2019

#1
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https://web2.0calc.com/questions/1-what-is-the-upper-bound-of-the-function

Sep 12, 2019
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Cphil, isnt here to help me since that link was from earlier but i beilieve the answer is actually 13

jjennylove  Sep 12, 2019
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do you have any ideas?

jjennylove  Sep 12, 2019
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Divide each term of the polynomial by 4

4x^4   - 25x^2  - 5x   - 13

___       _____   ___   ___      =   (1x^4) - ( 25/4)x^2  - (5/4)x - 13/4

4             4         4        4

Write down the coefficients    =    1     - 25/4        -5/4      -13/4

Drop the first one and remove the minus signs from any of the rest  ....we get ... 25/4     5/4     13/4

We have two  possible bounds on the polynomial

a)  Take the largest value of the above and add 1  =   25/4 + 1 =  25/4 + 4/4   = 29/4  = 7.25

b)  Take the larger of :  the sum of the coefficients or 1....Add the coefficients = 43/4.....this is greater than 1....so the second possible bound is  43/4  = 10.75

Select the smaller  of  (a)  or (b)  =  7.25

This tells us that the upper and  lower bounds on zeroes are  7.25   and - 7.25

So.....the upper bound  is x = 7.25.....in other words......no real root will be  > 7.25   Sep 12, 2019
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Is it possible for it to be 7 ? Because when I didive 7 by the polyonmial I get all postitive values and that is lower than 7.25?

jjennylove  Sep 12, 2019
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I don't quite understand what you mean.......we are looking for the possible upper bound on zeroes.....in other words.....we are trying to find the largest x value such that no positive root can exceed this upper bound

Look at the graph here  :  https://www.desmos.com/calculator/omunuy8c4t

Note that the largest real root occurs at about (2.68, 0).....so   this confirms the fact that we do not need to search for any real roots  > 7.25 because they do not exist   CPhill  Sep 12, 2019
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https://www.chegg.com/homework-help/explore-cases-0-upper-bound-lower-bound-real-zeros-polynomia-chapter-4.2-problem-69e-solution-9780073519517-exc

If you click on this link the upper boun therm states "by using synthetic divsion if alll numbers are postitive and then thats the upper bound "

When i solve 7 into the equation so 7/4 -25 -5 -13 , i got all postitive values and then once i do 6 it is a mis of pos and neg values

jjennylove  Sep 12, 2019
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what do you think about that ?

jjennylove  Sep 12, 2019