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**Unformatted text preview: **x . How large can f (2) possibly be? Example 5 (4.2) NOTE Care must be taken in applying the Theorem “If f ’( x ) = 0 for all x in an interval ( a , b ), then f is constant on ( a , b )”. s Let s The domain of f is D = { x | x ≠ 0} and f ’ ( x ) = 0 for all x in D . However, f is obviously not a constant function. This does not contradict the Theorem because D is not an interval. s Notice that f is constant on the interval (0, ∞ ) and also on the interval (-∞ , 0). 1 if ( ) 1 if | | x x f x x x > = = -< MEAN VALUE THEOREM Prove the identity tan-1 x + cot -1 x = π /2. s Although calculus isn’t needed to prove this identity, the proof using calculus is quite simple. Example 6 (4.2)...

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