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Do you know how to find derivatives?

yes..i do..

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

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

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

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

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

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

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

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

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

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

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

competitive*

You want the global or absolute minimum, correct?

try to factor it

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

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

Which gives -1543

@sauravshakya, there are errors in your factoring above.

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

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

not confirmed,,i mean obtained*

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