Determine Whether the Mean Value theorem can be applied to f on the closed inter

Question

Determine Whether the Mean Value theorem can be applied to f on the closed interval [a, b]. (select all that apply.) f(x) = squareroot 5 - x, [-20, 5] Yes, the Mean Value Theorem can be applied. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in the open interval (a, b). None of the above. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that f'(c) = f(b) - f(a)/b - a. (Enter your answers as a comma-separated list. If the Mean Value Theorem cannot be applied, enter NA.) c =

Explanation / Answer

Verify that f (x) = (5-x)^.5 satisfies the hypotheses of the Mean Value Theorem on [-20,5]. A theorem is guaranteed to hold only when all its hypotheses are satisfied.
f (x) is always continuous, so requirement 1 is met.
If f (x) is differentiable, that means f '(x) is continuous. f '(x) =-5(5-x)^-.5 which is always continuous, so requirement 2 is met.

Yes the mean value theorem can be applied

[f (b) – f (a)]/(b – a) = f '(c)

[f (-20) – f (5)]/(-20-5) = f '(c)

5/(-25)=5(5-c)^-.5

c=-620

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