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Show that if f is differentiable and f'(x) > 0 on (a, 6), then f is strictly increasing provided there is no subinterval (c, d) with c< d on which f' is identically zero.

Show that if f is differentiable and f'(x) > 0 on (a, 6), then f is strictly increasing provided there is no subinterval (c, d) with c< d on which f' is identically zero.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Theorem: Conditions for Strictly Increasing Functions** **Statement**: Show that if \( f \) is differentiable and \( f'(x) \geq 0 \) on the interval \((a, b)\), then \( f \) is strictly increasing provided there is no subinterval \((c, d)\) with \( c < d \) on which \( f' \) is identically zero. **Explanation**: This theorem addresses the relationship between the derivative of a function and its behavior over an interval. The condition \( f'(x) \geq 0 \) implies that the function does not decrease. However, this condition alone isn't sufficient to guarantee that \( f \) is strictly increasing; the function could be constant on some subinterval. The additional requirement that there is no subinterval \((c, d)\) where \( f' \equiv 0 \) ensures that \( f \) is not constant on any segment of \((a, b)\), thereby ensuring that \( f \) is strictly increasing over the entire interval. **Applications**: This result is useful in calculus and analysis to determine when a function behaves monotonically over a given interval, especially in optimization and in studying the qualitative behavior of functions. If you have a specific function or scenario in mind, this theorem provides a clear criterion to check whether the function is strictly increasing.
Transcribed Image Text:**Theorem: Conditions for Strictly Increasing Functions** **Statement**: Show that if \( f \) is differentiable and \( f'(x) \geq 0 \) on the interval \((a, b)\), then \( f \) is strictly increasing provided there is no subinterval \((c, d)\) with \( c < d \) on which \( f' \) is identically zero. **Explanation**: This theorem addresses the relationship between the derivative of a function and its behavior over an interval. The condition \( f'(x) \geq 0 \) implies that the function does not decrease. However, this condition alone isn't sufficient to guarantee that \( f \) is strictly increasing; the function could be constant on some subinterval. The additional requirement that there is no subinterval \((c, d)\) where \( f' \equiv 0 \) ensures that \( f \) is not constant on any segment of \((a, b)\), thereby ensuring that \( f \) is strictly increasing over the entire interval. **Applications**: This result is useful in calculus and analysis to determine when a function behaves monotonically over a given interval, especially in optimization and in studying the qualitative behavior of functions. If you have a specific function or scenario in mind, this theorem provides a clear criterion to check whether the function is strictly increasing.
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