A differentiable function satisfies f ( x ) ∈ [ 1 , 5 ] for every real x . Find the maximum possible value of f ( 3 ) − f ( 2 ) .

By the mean value theorem, f ( 3 ) − f ( 2 ) = f ′ ( c ) for some c ∈ ( 2 , 3 ) , so it is at most 5 . The bound is attained by f ( x ) = 5 x + C .

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MathematicsApplication of DerivativeMonotonicity-LMVTEasy2 minQB

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A differentiable function satisfies $f (x) \in [1, 5]$ for every real $x$ . Find the maximum possible value of $f (3) - f (2)$ .

Answer:

5.00

Solution:

By the mean value theorem, $f (3) - f (2) = f' (c)$ for some $c \in (2, 3)$ , so it is at most 5 . The bound is attained by $f (x) = 5 x + C$ .

Stream:JEESubject:MathematicsTopic:Application of DerivativeSubtopic:Monotonicity-LMVT

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