Does the function satisfy the hypothesis of the Mean Value Theorem on the given interval? f(x) = ln x, [1, 4] Does the function satisfy the hypothesis of the Mean Value Theorem on the given interval?...

Does the function satisfy the hypothesis of the Mean Value Theorem on the given interval? f(x) = ln x, [1, 4]

Does the function satisfy the hypothesis of the Mean Value Theorem on the given interval? f(x) = ln x, [1, 4]. If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem.

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The interval in question is `[a,b] = [1,4]`

The mean value theorem says that if a function is continuous on `[a,b]` (closed interval) and differentiable on `(a,b)` (open interval) then there is a point `c` in `(a,b)` such that

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

Now, we want `f'(c) = (ln(b) -ln(a))/(b-a) = ln(b/a)/(b-a) = ln(4)/3 = ln(4^(1/3))`

Using `ln(b) - ln(a) = ln(b/a)`  and `rlns = ln(s^r)`

We have `f(x) = ln x`

So we want `c` such that `lnc = ln(4^(1/3))`

This implies `c = 4^(1/3) = 1.587`

Yes, c = 1.587

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