Local Maxima and Minimum - No Work Required Only Logic

Explanation and answers are greatly appreciated along with any help. Thanks in advance!
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Assume that the polynomial P(x) has exactly two local maxima and one local minimum, and that these are the only critical points of P(x). Sketch possible graphs of P(x) and use them to answer the following.

(a) What is the largest number of zeros P(x) could have?

(b) What is the least number of zeros P(x) could have?

(c) What is the least number of inflection points P(x) could have?

(d) What is the smallest degree P(x) could have?

(e) What is the sign of the leading coefficient of P(x) ?
positive or negative?

(1) 4. That's when the maxima > 0 and minimum < 0.

(b) 2, when fmin > 0.

(c) 2, the function has at least two concave and one convex points

(d) 4. Degree >= max # of possible roots.

(e) -. Two local maxima --> P(+inf) < 0 --> the coefficients of highest term < 0.


[Ed. As the G mentioned, my answer to (b) is wrong. Actually, it should be zero, when both maxima sink below the x-axis.]

The general graph of P(x) would be either a parabola with the ends flattening out like

_........._
..\....../
...\__/
(i)
........... or simply a parabola
..\....../
...\__/

(ii)

both i and ii have two local maxima and one minima. The maximas do not have to be flat, just the highest points on the graph


a.) (i) has 3 zeros (the three flat parts or zeros)
b.) (ii) has only one zero( only one flat part)
c.) both i and ii have two inflection points - you can't have any less with two max's and one min
d.) the smallest degree is 2 ( they mean x raised to what? )
e.) positive ( if neg. the parabola would be upside down giving two mins and one max