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shubhamsrg Group Title

the minimum value of x^8 -8x^6 + 19x^4 -12x^3 +14x^2 -8x +9 = ?

  • 2 years ago
  • 2 years ago

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  1. CliffSedge Group Title
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    Do you know how to find derivatives?

    • 2 years ago
  2. shubhamsrg Group Title
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    yes..i do..

    • 2 years ago
  3. CliffSedge Group Title
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    You can try it that way, but of course that will involve solving a 7th degree polynomial...

    • 2 years ago
  4. shubhamsrg Group Title
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    ahh,,i'd have never thought of that! ;)

    • 2 years ago
  5. CliffSedge Group Title
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    I also recommend graphing it as you go to help narrow down where it might be.

    • 2 years ago
  6. shubhamsrg Group Title
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    well these solutions seem fine but are quite tedious,,dont you think ?

    • 2 years ago
  7. shubhamsrg Group Title
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    and afterall,,its no olympiad problem, just a 12th standard problem..so has so be a quick solution.,.

    • 2 years ago
  8. CliffSedge Group Title
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    Yes, but here's some good news, the minimum occurs as integer values of (x,y).

    • 2 years ago
  9. CliffSedge Group Title
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    ^ LOL, yeah, my TI-84 already solved this for me, but I like doing these things by hand too.

    • 2 years ago
  10. CliffSedge Group Title
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    It's actually a pretty quick one to find using rational root theorem and synthetic division.

    • 2 years ago
  11. CliffSedge Group Title
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    I'm also checking to see if the first derivative is factorable.

    • 2 years ago
  12. CliffSedge Group Title
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    Well, here's what I did: \[\large f'(x)=8x^7-48x^5+76x^3-36x^2+28x-8\] Factor out a 4 and set equal to zero. \[\large 2x^7−12x^5+19x^3−9x^2+7x−2=0\] Possible rational roots ε ± {1, 2, 1/2} Tested 1, didn't work, tested 2, found it.

    • 2 years ago
  13. CliffSedge Group Title
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    Followed by second derivative test, a little point-plotting to verify, and yeah, not too bad.

    • 2 years ago
  14. shubhamsrg Group Title
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    i'd still look forward to a shorter solution ! ..though i completely appreciate your method..

    • 2 years ago
  15. CliffSedge Group Title
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    Is this for a calculus course or algebra/pre-calc?

    • 2 years ago
  16. CliffSedge Group Title
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    Have you seen similar examples from your teacher or in your book? What tools are you expected to use? Are you required to do it by hand, or are graphing calculators allowed? All these considerations will determine the best way to do it.

    • 2 years ago
  17. shubhamsrg Group Title
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    am not doing any school work sir..am preparing for ISI .. the question is in its practice paper.. treat is a 12+ competetice exam..

    • 2 years ago
  18. shubhamsrg Group Title
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    competitive*

    • 2 years ago
  19. calculusfunctions Group Title
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    You want the global or absolute minimum, correct?

    • 2 years ago
  20. mukushla Group Title
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    try to factor it

    • 2 years ago
  21. mukushla Group Title
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    \[(x^8 -8x^6 + 19x^4 -12x^3 +14x^2 -8x +8)+1\]

    • 2 years ago
  22. sauravshakya Group Title
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    x^8 -8x^6 + 19x^4 -12x^3 +14x^2 -8x +9 x^8+19x^4-(8x^6 +12x^3) +(14x^2-8x)+9 (x^4-19/2)^2 +14 (x^2-4)^2-8(x^3+6)^2-17.25

    • 2 years ago
  23. sauravshakya Group Title
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    So, minimum when x^2-4=0 Thus, x=+-2

    • 2 years ago
  24. sauravshakya Group Title
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    Which gives -1543

    • 2 years ago
  25. calculusfunctions Group Title
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    @sauravshakya, there are errors in your factoring above.

    • 2 years ago
  26. estudier Group Title
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    http://www.willamette.edu/~mjaneba/help/Polynomial%20root%20finding.html Frankly, the purpose of this sort of thing (ie large degree polynomial by hand) can only be just the practice of math skills and to help remember all those bits and pieces about polynomials.... And, since it has been set as a question, you can assume there is a (relatively) easy solution of some sort.

    • 2 years ago
  27. shubhamsrg Group Title
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    ans. is 1 i tired factoring as @mukushla said.. i got somewhere but not enough progress..

    • 2 years ago
  28. CliffSedge Group Title
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    Did you try the first and second derivative tests like I did? It was pretty easy to track down the solution that way.

    • 2 years ago
  29. shubhamsrg Group Title
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    indeed,,your solution yielded the correct ans,,i.e. minima at x=2 .. and i appreciate that.. but still looking for shorter and sweeter solution..

    • 2 years ago
  30. CliffSedge Group Title
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    calc-minimum function on a graphing calculator? I don't know anything shorter or sweeter than that. ;-)

    • 2 years ago
  31. estudier Group Title
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    I doubt there is anything faster by hand than what Cliffsedge has given already....

    • 2 years ago
  32. shubhamsrg Group Title
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    i was able to check that(following @mukushla 's work) x=2 was a root of f(x) - 1 which can be confirmed from rational root theorem,, will that be satisfactory??

    • 2 years ago
  33. shubhamsrg Group Title
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    not confirmed,,i mean obtained*

    • 2 years ago
  34. estudier Group Title
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    Well, that's about as good as making an inspired guess that there is a root of 2. If you want to go down that route, then say to yourself well, the x^8 obviously takes over for x greater than plus or minus 3, so test 0,1,2 and voila! you found it....

    • 2 years ago
  35. CliffSedge Group Title
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    I agree, finding the root of f(x)-1 seems like too much of a lucky guess.

    • 2 years ago
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