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

- shubhamsrg

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

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- anonymous

Do you know how to find derivatives?

- shubhamsrg

yes..i do..

- anonymous

You can try it that way, but of course that will involve solving a 7th degree polynomial...

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- shubhamsrg

ahh,,i'd have never thought of that! ;)

- anonymous

I also recommend graphing it as you go to help narrow down where it might be.

- shubhamsrg

well these solutions seem fine but are quite tedious,,dont you think ?

- shubhamsrg

and afterall,,its no olympiad problem, just a 12th standard problem..so has so be a quick solution.,.

- anonymous

Yes, but here's some good news, the minimum occurs as integer values of (x,y).

- anonymous

^ LOL, yeah, my TI-84 already solved this for me, but I like doing these things by hand too.

- anonymous

It's actually a pretty quick one to find using rational root theorem and synthetic division.

- anonymous

I'm also checking to see if the first derivative is factorable.

- anonymous

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.

- anonymous

Followed by second derivative test, a little point-plotting to verify, and yeah, not too bad.

- shubhamsrg

i'd still look forward to a shorter solution ! ..though i completely appreciate your method..

- anonymous

Is this for a calculus course or algebra/pre-calc?

- anonymous

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.

- shubhamsrg

am not doing any school work sir..am preparing for ISI .. the question is in its practice paper..
treat is a 12+ competetice exam..

- shubhamsrg

competitive*

- calculusfunctions

You want the global or absolute minimum, correct?

- anonymous

try to factor it

- anonymous

\[(x^8 -8x^6 + 19x^4 -12x^3 +14x^2 -8x +8)+1\]

- anonymous

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

- anonymous

So, minimum when x^2-4=0
Thus, x=+-2

- anonymous

Which gives -1543

- calculusfunctions

@sauravshakya, there are errors in your factoring above.

- anonymous

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.

- shubhamsrg

ans. is 1
i tired factoring as @mukushla said..
i got somewhere but not enough progress..

- anonymous

Did you try the first and second derivative tests like I did?
It was pretty easy to track down the solution that way.

- shubhamsrg

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..

- anonymous

calc-minimum function on a graphing calculator?
I don't know anything shorter or sweeter than that.
;-)

- anonymous

I doubt there is anything faster by hand than what Cliffsedge has given already....

- shubhamsrg

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??

- shubhamsrg

not confirmed,,i mean obtained*

- anonymous

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....

- anonymous

I agree, finding the root of f(x)-1 seems like too much of a lucky guess.

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