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1.)If f is continuous on (−∞, ∞), what can you say about its graph? (Select all that apply.)

a)The graph of f has a hole.

b)The graph of f has a jump.

c)The graph of f has a vertical asymptote.

d)none of these

10.)

EXAMPLE 10 Show that there is a root of the equation

4x³− 6x² + 3x − 2 = 0

between 1 and 2.

SOLUTION Let

f(x) = 4x³− 6x² + 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

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 and such that

f(c) =  .

In other words, the equation

4x³− 6x² + 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(1.2) = −0.128 < 0    and    f(1.3) = 0.548 > 0

a root must lie between (smaller) and (larger). A calculator gives, by trial and error,

f(1.22) = −0.007008 < 0    and    f(1.23) = 0.056068 > 0.

So a root lies in the open interval

.