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Published byNorah Barrett Modified over 6 years ago

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Why is it the second most important theorem in calculus?

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Two closely related facts: suppose we have some fixed constant C and differentiable functions f and g. If f (x) C, then f ’ (x) 0. If f (x) g(x) + C, then f ’ (x) g ’ (x). Suppose we have a differentiable function f. If f is increasing on (a,b), then f ’ 0 on (a,b). If f is decreasing on (a,b), then f ’ 0 on (a,b). How do we prove these things?

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Let’s try one!

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Two closely related facts: suppose we have some fixed constant C and differentiable functions f and g. If f ’ (x) 0, then f (x) C. If f ’ (x) g ’ (x), then f (x) g(x) + C. Suppose we have a differentiable function f. If f ’ 0 on (a,b), then f is increasing on (a,b). If f ’ 0 on (a,b), then f is decreasing on (a,b). How do we prove these things?

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Proving these requires more “finesse.”

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