Question

the minimum value of

x^8 -8x^6 + 19x^4 -12x^3 +14x^2 -8x +9 = ?

Answer 1

Do you know how to find derivatives?

Answer 2

yes..i do..

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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.

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

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.

Answer 17

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

Answer 18

competitive*

Answer 19

You want the global or absolute minimum, correct?

Answer 20

try to factor it

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

Which gives -1543

Answer 25

@sauravshakya, there are errors in your factoring above.

Answer 26

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.

Answer 27

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

Answer 28

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

Answer 29

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..

Answer 30

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

Answer 31

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

Answer 32

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??

Answer 33

not confirmed,,i mean obtained*

Answer 34

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....

Answer 35

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