Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. f(x) = x^7, [0, 7]

GIVEN: The given function is f(x)= on the interval [0,7]

To FIND: Here we need to find the value of c by the help of Mean value theorem.

SOLUTION: The mean value theorem is,

where, f(x) is defined on a closed interval [0,7] and continuous on the closed [0,7] and derivable on (0,7) and a<c<b,

So,

By the mean value theorem we have,

Therefore, the required value of c is 5.077