Determine whether the Mean Value Theorem applies 9. Determine whether the Mean V

Question

Determine whether the Mean Value Theorem applies

9. Determine whether the Mean Value Theorem applies to f(x) = -1/x on the intevall [-3, -1/2], If the Mean Value Theorem applies, find all values of c in the interval such that P f?(c) = f(-1/2)-f(-3)/-1/2-(-3) . If the Mean Value Theorem does not apply, state why. (a) Mean Value Theorem applies; c = - squareroot 3/2. (b) Mean Value Theorem applies, c = squareroot 3/2. (c) The Mean Value Theorem does not apply because f is not continuous at x = 0. (d) The Mean Value Theorem does not apply because f (-1/2) not equal to f(-3). (e) None of these

Explanation / Answer

f(x)= -1/x is continuous on [-3, -1/2] and also differentiable on (-3, -1/2). So Mean Value Theorem is applicable here.

By meean value theorem there exist a point c in (-3, -1/2) such that f'(c) = {f(-1/2) - f(-3)} / {-1/2 - (-3)}

or, f'(c) = (2-1/3) / (5/2) = 2/3   

Now f'(x) = 1/(x^2)

So 1/(c^2) = 2/3 ,   or, c^2 = 3/2 ,   or, c = +sqrt(3/2) , -sqrt(3/2)   

But, +sqrt(3/2) is not in the interval (-3, -1/2).

So, c = -sqrt(3/2)  

Hence the option (a) is the correct.

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