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72 2. Evaluation, Division, and Expansion

Let f(z) be a polynomial defined on R. We say that f(x) is increasing
on an interval [a, b] = {x : a 5 x 5 b} if, for each pair of values u, v
within the interval such that u 5 v, f(u) 5 f(v). It can be shown that f is
increasing on the interval if and only if its derivative is nonnegative there.
f(x) is decreasing on [a, b], if, for a 2 TJ < v 5 b, f(u) 1 f(v). This is
equivalent to asserting that its derivative is nonpositive on the interval.
f(x) has a maximum at the point c if there is some small interval with
c in its interior such that f(x) 5 f(c) whenever x lies in the interval.
Necessarily, at each maximum, f’(c) = 0.
f(x) has a minimum at the point c if on some small interval with interior
point c, f(x) 2 f(c) f or each x in the interval. Again, this implies that
f’(c) = 0.
c is a critical point for f if f’(c) = 0. At a critical point, f could have a
maximum, a minimum or neither a maximum nor a minimum.
We will use the fact that each polynomial f(x) is continuous in 2. This
means that small changes in the value of x give rise to small changes in
f(x). Thus, a graph of a polynomial is a smooth curve without any breaks
or corners. One consequence of this is that somewhere on each interval
[a,b], f(x) assumes every value which lies between f(a) and f(b).

Exercises

1. Sketch the graph of a typical constant polynomial.


2. Sketch the graph of a polynomial of the form ax + b, where a and b
   are real with a nonzero. Deduce from the graph that this polynomial
   assumes each real value exactly once.


3. (4
   Q-4

(4

(4

Sketch the graphs of the polynomials x2 and (x - k)‘.

Using the representation

ax2+bx+c=a(x+$)2- (y)

sketch the graph of the polynomial ax2 + bx + c, where a, b,

c are real and a is nonzero. Distinguish the cases in which the

leading coefficient and the discriminant are separately positive

and negative. Determine on the graph all maxima and minima

for the function.

On the same axes as in (b), sketch the graph of the derivative

of the quadratic. Relate the values of the derivative to the be-

haviour of the quadratic function.

Deduce that no polynomial over R of degree 2 can assume all

possible real values.