IXL - Mean Value Theorem (Calculus practice)

Does the function

satisfy the hypotheses of the Mean Value Theorem on the interval

[

]

If so, what is the conclusion of the Mean Value Theorem? If not, why not?

(

)

Yes. By the Mean Value Theorem, there is a number

(

)

such that

′

(

)

Yes. By the Mean Value Theorem, there is a number

(

)

such that

′

(

)

No.

is continuous on

[

]

but

is not differentiable at

No.

is not continuous at

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Learn with an example

question

Does the function

satisfy the hypotheses of the Mean Value Theorem on the interval

[

]

If so, what is the conclusion of the Mean Value Theorem? If not, why not?

(

)

Yes. By the Mean Value Theorem, there is a number

(

)

such that

′

(

)

Yes. By the Mean Value Theorem, there is a number

(

)

such that

′

(

)

No.

is continuous on

[

]

but

is not differentiable at

No.

is not continuous at

key idea

The Mean Value Theorem states that if a function

continuous on the closed interval
[
a
,
b
]
and
differentiable on the open interval
(
a
,
b
)
,

then there is at least one number

(

)

such that

′

(

)

(

)

–

(

)

–

solution

Determine if the function

satisfies the hypotheses of the Mean Value Theorem on the interval

[

]

The function

(

)

is a rational function, so it is continuous on its entire domain, which is the set of all real numbers except for

–

So,

is continuous on the interval

[

]

Since

(

)

is a rational function, it is differentiable on its entire domain, which is the set of all real numbers except for

–

and

′

(

)

′

Use the quotient rule

(

)

(

)

′

–

(

)

(

)

′

(

)

Use the quotient rule

(

)

(

)

–

(

)

(

)

(

)

(

)

So,

is differentiable on the interval

(

)

Therefore,

does satisfy the hypotheses of the Mean Value Theorem on the interval

[

]

This means that there is at least one number

(

)

such that

′

(

)

(

)

–

(

)

–

′

(

)

–

′

(

)

–

(

)

′

(

)

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H.9Find derivatives using the product rule I

H.11Find derivatives using the quotient rule I

H.14Find derivatives using the chain rule I

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