EXAMPLE 10: Show that there is a root of the equation
8x^3 - 10x^2 + 3x - 2 = 0
between 1 and 2.
SOLUTION:
Let f(x) = 8x^3 - 10x^2 + 3x - 2 = 0.
We are looking for a solution of the given equation, that is, a
number c between 1 and 2 such that
f(c) = ________. Therefore we take a = _________, b = _________,
and N = _________ in the Intermediate Value Theorem. We have
f(1) = 8 - 10 + 3 - 2 = -1 < 0
and
f(2) = 64 - 40 + 6 - 2 = 28 > 0
Thus f(1) < 0 < f(2); that is N = 0 is a number between
f(1) and f(2). Now f is continuous since it is a polynomial, so
that the Intermediate Value Theorem says there is a number c
between 1 and 2 such that f(c) = _______. In other words, the
equation 8x^3 - 10x^2 + 3x - 2 = 0 has at least one root c in the open interval
_______.
In fact, we can locate a root more precisely by using the
Intermediate Value Theorem again. Since
f(1.1) = -0.152 < 0 and f(1.2) = 1.024 > 0
a root must lie between ______ (smaller) and ________ (larger).
A calculator gives, by trial and error,
f(1.11) = -0.049952 < 0 and f(1.12) = 0.055424 > 0
so a root lies in the open interval _______.