Chapter 4 | Applications of Derivatives
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Figure 4.29 If a function has a positive derivative over some interval I , then the function increases over that interval I ; if the derivative is negative over some interval I , then the function decreases over that interval I .
Theorem 4.8: Corollary 3: Increasing and Decreasing Functions Let f be continuous over the closed interval [ a , b ] and differentiable over the open interval ( a , b ).
i. If f ′( x ) >0 for all x ∈ ( a , b ), then f is an increasing function over [ a , b ]. ii. If f ′( x ) <0 for all x ∈ ( a , b ), then f is a decreasing function over [ a , b ].
Proof We will prove i.; the proof of ii. is similar. Suppose f is not an increasing function on I . Then there exist a and b in I such that a < b , but f ( a ) ≥ f ( b ). Since f is a differentiable function over I , by the Mean Value Theorem there exists c ∈ ( a , b ) such that f ′( c ) = f ( b )− f ( a ) b − a . Since f ( a ) ≥ f ( b ), we know that f ( b )− f ( a ) ≤0. Also, a < b tells us that b − a >0. We conclude that f ′( c ) = f ( b )− f ( a ) b − a ≤0. However, f ′( x ) >0 for all x ∈ I . This is a contradiction, and therefore f must be an increasing function over I . □
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