Please show work!

Question

Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. (Round your answer to four decimal places. Enter your answers as a comma-separated list.)

f(x) = 7x [4,9]

Answer 1

The Mean Value Theorem for integrals states that a function f that is continuous over the interval [a,b] has a value of c within that interval such that f'(c) = [f(b)-f(a)]/(b-a)

Given our function is f(x)=7√x over the interval [4,9] we can find the average rate of change:

f'(c) = [f(b)-f(a)]/(b-a)

f'(c) = [f(9)-f(4)]/(9-4)

f'(c) = (21-14)/5

f'(c) = 7/5

Because f'(c) = 7/(2√c), we can set this equal to 7/5 from before and solve for c:

7/(2√c) = 7/5

2√c = 5

√c = 5/2

c = 25/4

Hence, the value of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval is c = 25/4, or c = 6.2

Hope this helped!

Answer 2

f(x) = 7sqrx from x=4 to 9

f(9) = 7sqr9 = 21

f(4) = 7sqr4 = 14

(21-14)/(9-4) = 7/5

f'(x) = (7/2)/sqrx

f'(c) = 3.5/sqrc = 1.4

sqrc = 5/2

c = 6.25

Answer 3

f '(c) = (f(b) - f(a))/(b-a)

f(9) = 21

f(4)=14

f(9)-f(4) = 7

b-a = 9-4 = 5

f '(c) = 7/(2√c) = 7/5

Solve for c