GIVEN: The given function is f(x)=[tex]x^7[/tex] 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,
[tex]f'(c)=\frac{f(b)-f(a)}{b-a}[/tex]
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, [tex]f'(x)=7x^6[/tex]
By the mean value theorem we have,
[tex]7c^6=\frac{7^7-0}{7-0} \\7c^6=7^6\\c^6=7^5\\c=\sqrt[6]{7^5} \\c=5.077[/tex]
Therefore, the required value of c is 5.077
