Is there any body who can help me with math.?

Question

1. Given a 4th degree polynomial with a negative leading coefficient and positive constant term:

a.Discuss the possible number of real zeroes, counting multiplicity.
b.Discuss the possible number of local maxima
c.Discuss the possible number of local minima
d.Discuss the absolute maximum or minimum point(s)
e.Discuss the end behavior.

Answer 1

a.Discuss the possible number of real zeroes, counting multiplicity.
it has to have at least 1 zero. The possibilities are the following:
- 1 zero of multiplicity 1
- 2 zeroes for multiplicyt 2 each
- 3 zeroes, one with multiplicyt 2 and the rest with multiplicity 1
- 4 zeroes with multiplicity 1.
b.Discuss the possible number of local maxima
- it has at least 1 local maxima.
-it may have 2 local maxima
c.Discuss the possible number of local minima
- at most 1 local minima
d.Discuss the absolute maximum or minimum point(s)
it has to have 1 absolute maximum
and no absolute minimum.
e.Discuss the end behavior.
lim x-> infinity f(x) = - infinity
lim x-> - infinity f(x) = - infinity
where f(x) is the polynomial you describe in your problem.

Answer 2

a) 4 possible real zeros

b) two possible local maxima because the negative leading coefficient means the ends point to -infinity. That being said if there is a possibility for 4 real zeros, you could possibly go neg. to pos., pos to neg., neg. to pos. and pos. to neg. (in terms of y) So that leaves room for 2 local maxima because y goes neg. to pos. then pos. to neg. twice.

c) 4 possible roots implies 3 local max/min since 2 can be local maxima, there is one local minima

d) absolute maximum will occur at one of the local maxima.

e) ends will tend to -infinity