2.4. Graphing Polynomials 73
(e) Show that, if a quadratic polynomial has two real roots, then its
derivative must have its root between them.- Sketch the graphs of the following cubits. For each, on the same axes,
sketch the graphs of its first and second derivatives.
(4 x3
(b) x3+ 8
(c) x3- 8
(d) x3 + 12x
(e) x3 - 12x
(f) x3 + ax; distinguish the cases that a is positive and negative
(g) x3 + ax + b.- Consider the general cubic over R, ax3 + bx2 + cx + d. Show that
there is a change of variable of the form s = x + k, which will render
it in the form as3 + ms + n for some real numbers m and n. Use this
fact to discuss the graph of the general cubic. - Let f(t) be a cubic polynomial. The point u in R at which fâ van-
ishes is called an inflection point. The point (u, f (u)) on the graph of
the cubic generally separates the convex part of the graph from the
concave part. Show that the graph of any cubic is centrally symmetric
about its inflection point. (You have to show that, if (u-v, f(u) - w)
is on the graph, so also is the point (u + v, f(u) + w). - Use the results of your investigation on cubits to argue that
(a) every cubic with real coefficients has at least one real zero;
(b) every cubic with real coefficients assumes every real value at
least once.- Sketch the graphs of the following quartics. On the same axes for
each, sketch its first and second derivatives.
(b) x4 + 3x2 + 2
(c) x4 - 3x2 + 2
(d) x4 - 5x - 6
(e) x4 + 5x - 6
(f) x4 + 3x2 - 36x
(g) x4 + 3x2 - 36x + 48