Show me how this equation is valid

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SUMMARY

The discussion centers on the validity of the equation ##f \geq 0## and its implications for the inequality ##y \geq x \implies yf(y) \geq xf(y)##. Participants confirm that since y encompasses all values greater than or equal to x, the constant x is effectively substituted with the increasing function y within the integral. This substitution is crucial for understanding the behavior of the function in the context of the inequality.

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  • Understanding of basic calculus concepts, particularly integrals.
  • Familiarity with inequalities and their properties in mathematical analysis.
  • Knowledge of functions and their behavior, specifically increasing functions.
  • Basic comprehension of mathematical notation and symbols used in inequalities.
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  • Study the properties of increasing functions and their implications in calculus.
  • Explore the concept of integrals and how substitutions affect their evaluation.
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  • Investigate the role of non-negative functions in integrals and inequalities.
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Mathematics students, educators, and anyone interested in deepening their understanding of inequalities and integrals in calculus.

oahz
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Please, hurry. It's driving me nuts
 

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I suppose ##f \geq 0##? If so, ##y\geq x \implies yf(y) \geq xf(y)##.
 
Seconded. Note that y takes on every value >= x, so you have basically replaced the constant x with the increasing function y in the integral.
 

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