Only one option is correct ,Please help ; How to solve this question?

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1 Answer
May 10, 2017

D.

Explanation:

f is differentiable on each of (0,2) and (4,6).
Since differentiability implies continuity, f is also continuous on each of the open intervals.

So, for any interval [a,b] entirely within one of these open intervals, we can apply the Mean Value Theorem

f is continuous on [a,b] and
f is differentiable on (a,b)

Therefore, there is a c in (a,b) such that

f(b)-f(a) = f'(c)(b-a).

Furthermore, since f'(c) = 1, we see that

So, for any interval [a,b] entirely within one of these open intervals,

f(b)-f(a) = (b-a).

Therefore,

f(5.5)-f(4.5) = 1 = f(1.5)-f(0.5)

The function

f(x) = {(x+2,"if",0 < x < 2),(x-2,"if",4 < x < 6):} satisfies the given condition but fails A, B, and C. The graph is shown below.

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