$\"\"$

The function is continuous and $\"\"$ .

Mean value theorem for integrals:

If $\"\"$ is continuous on $\"\"$ , then there exist a number $\"\"$ in $\"\"$ such that $\"\"$ .

Since the function is continuous on $\"\"$ , so by the mean value theorem, there exist a number $\"\"$ in $\"\"$ .

Such that

$\"\"$ .

We have, $\"\"$ .

Substitue $\"\"$ in $\"\"$ .

$\"\"$ .

From the above expression it is observed that there exist a number $\"\"$ in $\"\"$

Such that $\"\"$ .

Therefore, the function value takes on the value $\"\"$ on $\"\"$ .

$\"\"$

The function value takes on the value $\"\"$ atleast once on $\"\"$ .