Mean Value Theorem for Integrals Problem

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I'm having difficulty getting a proper answer for this problem. I'm not that good at Mean Value Theorem for Integrals. Can someone help me find the solution?

Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval.

$f$($x$) = $x^5$, [0, 5]

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2 Answers

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The mean value theorem for integrals says that there exists a $c$ for which $f$ agrees with its average; that is,

$$f(c) = \frac1{5 - 0} \int_0^5 f(t) dt$$

So compute the integral

$$\int_0^5 t^5 dt$$ and get a number; you should get $5^6/6$; then solve

$$c^5 = \frac{5^5}{6}$$

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This is the answer I got when I solved it. I hope you understand it.

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