Name: Intermediate Mean Value Theorem
Uploaded: 2019-07-19
Duration: 13 min 11 s
Description: Let's discuss the important theorem based on the continuity of the function, known as the intermediate mean value theorem. Watch this video to learn the conc...

Question

If

f (x)

is continuous in

[a, b]

such that

f (a) = b

and

f (b) = a,

then prove that there exists at least one

c \in (a, b)

such that

f (c) = c

.

Accepted Answer

Given, $f (x)$ is continuous function in $[a, b]$ and $f (a) = b, f (b) = a$

Then for at least one $c \in (a, b)$ there should be $f (c) = c$

Let $g (x) = f (x) - c$

Given, $f (x)$ is continuous in $[a, b]$ and we know that constant function is always continuous

So $g (x)$ is also continuous in $[a, b]$

Now, $g (a) = f (a) - c = b - c > 0 (∵ a < c < b)$

$g (b) = f (b) - c = a - c < 0 (∵ a < c < b)$

So we can say that in between $a & b$ there will be at least one root of $g (x)$

So for at least one $c \in (a, b)$

$g (c) = 0$

$\Rightarrow f (c) - c = 0$

$\Rightarrow f (c) = c$

If $f (x)$ is continuous in $[a, b]$ such that $f (a) = b$ and $f (b) = a,$ then prove that there exists at least one $c \in (a, b)$ such that $f (c) = c$ .

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

If fx is continuous in a,b such that fa=b and fb=a, then prove that there exists at least one c∈a,b such that fc=c.

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

If $f (x)$ is continuous in $[a, b]$ such that $f (a) = b$ and $f (b) = a,$ then prove that there exists at least one $c \in (a, b)$ such that $f (c) = c$ .