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

Question

f(x) = x^4, [0, 4]

Answer 1

Step 1: multiply one over b-a by the integral of f(x) from [a,b]
1 b
--------- ∫ f(x) dx
b - a a

1 4
--------- ∫ x^4 dx
4 - 0 0

1 [ x^5 ] 4
----- [ -------- ]
4 [ 5 ] 0

1 [ (4)^5 (0)^5 ]
----- [ --------- -- ---------- ]
4 [ 5 5 ]

1 [ 1024 ] 256
----- [ -------- ] = --------
4 [ 5 ] 5

Answer 2

Step 2: plug in for f(c)

f(c) = x^4

256/5 to the 4th root

The answer is 4 over the 4th root of 5