True or False: A polynomial of odd degree which is defined for all (x) can have no absolute minimum..........?

Answer 1

True. As x goes to negative infinity, x raised to any odd power also goes to negative infinity. As x goes to positive infinity, x raised to any odd power also goes to positive infinity. The leading term of any polynomial function eventually dwarfs the other terms, so the extremes of the graph on the left and right are going to go to negative and positive infinity respectively. Therefore, while it may have local maxes and mins in the middle, it won't have any global maxes or mins.

Answer 2

For those respondants who said that the limit of such a polynomial as x -> -infinity is -infinity and the limit as x-> +infinity is +infinity, you are assuming that the leading coefficient is positive. The last respondant had the right idea, but the same idea carries over to polynomials of odd degree higher than 3. If the polynomial has degree n, then:

Case 1: If the leading term is ax^n with a >0, then the limit of the function as x-> + infinity is +infinity, and the limit as x->-infinity is -infinity.

Case 2: If the leading term is ax^n with a<0, then the limit of the function as x->+infinity is -infinity, and the limit as x->-infinity is +infinity.

In either case, given any y value, one can always find points on the graph with lower y-coordinates by going out sufficiently far out in some direction on the x-axis, so such functions cannot have an absolute minimum.

Answer 3

My first instinct is to say true, but give me a minute to find a proof.
Check out the website in the source. It looks to be of a beginning to intermediate level and will tell you more than you ever wanted to know about polynomials. The second source is also very good an even more basic.

Answer 4

True, because the limits of any odd-degree polynomal approach both +infinity and -infinity in two different directions. If the coefficient of x^3 is positive, then:

f(x) -> +inf when x -> +inf
f(x) -> -inf when x -> -inf

If the coefficient of x^3 is negative, then:

f(x) -> -inf when x-> +inf
f(x) -> +inf when x-> -inf

Answer 5

true.
such a polynomial goes to -INF when x goes to -INF
and to + INF when x goes to INF