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4 votes
F(x)=x^3- X-1; between 1 and 2

1 Answer

2 votes

Answer:

Here's one way to do it.

Step-by-step explanation:

Let

f

(

x

)

=

x

3

2

x

2

+

3

x

.

(Needed because the intermediate value theorem is a theorem about functions .)

Observe that the equation

x

3

2

x

2

+

3

x

=

5

has a root (a solution) exactly when

f

(

x

)

=

5

So the question now is to show that for at least one number

c

, in

[

1

,

2

]

, we get

f

(

c

)

=

5

.

f

is continuous on

[

1

,

2

]

(Because it is a polynomial and they are continuous everywhere.)

f

(

1

)

=

2

and

f

(

2

)

=

6

5

is between

f

(

1

)

and

f

(

2

)

, so

by the intermediate value theorem, there is at least one number

c

, in

[

1

,

2

]

, for which

f

(

c

)

=

5

.

That is, the original equation has a solution.

Step-by-step explanation:

User Uray
by
7.5k points

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