SUMMARY
The discussion centers on proving that a function \( f : \mathbb{R} \to (0, \infty) \) with the property \( f'(x) = f(x) \) for all \( x \) is an increasing function. It is established that if \( f'(x) > 0 \) for all \( x \) in an interval, then \( f \) is strictly increasing on that interval. The conclusion drawn is that since \( f'(x) = f(x) \) and \( f(x) > 0 \), it follows that \( f \) is indeed increasing for all \( x \).
PREREQUISITES
- Understanding of calculus, specifically derivatives and their implications on function behavior.
- Knowledge of the Mean Value Theorem and its application in proving function properties.
- Familiarity with the concept of exponential functions and their derivatives.
- Basic understanding of the real number system and intervals.
NEXT STEPS
- Study the properties of exponential functions, particularly \( e^x \) and its derivative.
- Learn about the Mean Value Theorem and how it applies to proving monotonicity of functions.
- Explore differential equations, focusing on first-order linear equations like \( f' = f \).
- Investigate the implications of \( f'(x) = f(x) \) in various contexts, including growth models in mathematics and biology.
USEFUL FOR
Students studying calculus, mathematicians interested in function analysis, and educators teaching concepts of derivatives and increasing functions.