Determining range of polynomial

Question

How do I find the range, max, and min values of f(X)=13x-20x-x^2-3x^4

Answer 1

set derivative equal to 0 and get critical points.

Answer 2

The "13x - 20x" part makes me think that you may have mistyped it, as I see no reason why they wouldn't be combined.  Ignoring that - this is a fourth-degree polynomial.  Since the highest order term is negative, it means it will open downward.  Consequently, there is no minimum.  It will rise to a maximum, descend, possibly rise to another maximum, then return back to -inf.  So - no minimum value, and the range will extend to minus infinity.

As for the maximum, if you know calculus, then your first answer is the best approach.  Find the critical points.  There will be three.  One will be a relative minimum, the other two will either be two absolute maxima or one relative maximum and one absolute maximum.  The value of the absolute maximum will give you the max value as well as the top of the range.

If you don't know calculus, you have more work to do.  Begin by factoring it (assuming that, as you're typed it, every term has an x in it).  If you can find the roots using Descartes' Rule of Signs and the Rational Root Theorem, so much to the good.  That will help you pin down where the maxima and minima are.  Either way, without calculus, I don't see any way to find those values except by trial and error.  Make a table and start calculating values.  You should see it rise, turn down, rise again, and finally turn down forever.  Once you know roughly where the turns come, you can pin them down further by calculating values closer and closer together.

I'd give you some help figuring out where you should look, but as I said I'm not sure you've typed the polynomial correctly.  Check it.  If it's right, post it in details.  If not, post the correct one there.  If I see it, I'll give you a range of values to check.