How do you verify the intermediate value theorem over the interval [5/2,4], and find the c that is guaranteed by the theorem such that f(c)=6 where $f(x)=(x^2+x)/(x-1)$ ?

Question

How do you verify the intermediate value theorem over the interval [5/2,4], and find the c that is guaranteed by the theorem such that f(c)=6 where $f(x)=(x^2+x)/(x-1)$ ?

1 Answer

Impact of this question

10082 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-20T23:51:53

Please see the verification in the explanation section.

Explanation:

$f$ is continuous on $[5/2,4]$

Proof : $f$ is a rational function and rational functions are continuous on their domains. $f$ is continuous at all reals except $1$ . (Its domain is all reals except $1$ .) Since $1$ is not is $[5/2,4]$ , $f$ is continuous on that closed interval.

$6$ is between $f(5/2)$ and #f(4)

Proof: $f(5/2) = 35/6 < 36/6 = 6$ and $f(4) = 20/3 > 18/3 = 6$

Therefore, there is a $c$ in $(5/2,4)$ with $f(c) = 6$

To find the $c$ (or $c$ 's), solve the equation:

$f(x) = 6$ .

Discard values outside $(5/2,4)$ .

Note the solutions are $2$ and $3$ , but only $3$ is in the interval.

Science

Math

Humanities

... and beyond

How do you verify the intermediate value theorem over the interval [5/2,4], and find the c that is guaranteed by the theorem such that f(c)=6 where $f(x)=(x^2+x)/(x-1)$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you verify the intermediate value theorem over the interval [5/2,4], and find the c that is guaranteed by the theorem such that f(c)=6 where f(x)=(x^2+x)/(x-1)f(x)=(x^2+x)/(x-1)?

1 Answer

Explanation:

Related questions

Impact of this question

How do you verify the intermediate value theorem over the interval [5/2,4], and find the c that is guaranteed by the theorem such that f(c)=6 where $f(x)=(x^2+x)/(x-1)$ ?