Question

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x) = x , [0, 4]

Answer 1

The number c is 2.

Explanation:

Mean Value Theorem:

If f is a continuous function in a bounded interval [0,4], there is at least one value of c in (a,b) for which:

`f(c) = (1)/(b-a)\int\limits^a_b {f(x)} \, dx`

In this problem, we have that:

`f(x) = x, a = 0, b = 4`

So
`f(c) = c`

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`f(c) = (1)/(b-a)\int\limits^a_b {f(x)} \, dx`

`c = (1)/(4-0)\int\limits^0_4 {x} \, dx`

`c = (1)/(4-0)*8`

`c = 2`

The number c is 2.