Derivative of monotone increasing and bounded f

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SUMMARY

The discussion centers on the behavior of the derivative of a monotone increasing, bounded, and differentiable function f on the interval (a, ∞). It is established that as x approaches infinity, the limit of the derivative, lim(f'(x), x, ∞), indeed approaches zero. This conclusion is drawn from the property that a bounded function must flatten out as it approaches its supremum, leading to smaller slopes of the tangent lines, which confirms that the derivative diminishes to zero.

PREREQUISITES
  • Understanding of monotone functions
  • Knowledge of bounded functions
  • Familiarity with derivatives and differentiability
  • Concept of limits in calculus
NEXT STEPS
  • Study the properties of monotone functions in calculus
  • Explore the concept of boundedness and its implications on function behavior
  • Learn about the relationship between derivatives and the behavior of functions at infinity
  • Investigate counterexamples for limits of derivatives in different function classes
USEFUL FOR

Mathematicians, calculus students, and anyone interested in the properties of monotone and bounded functions, particularly in the context of derivatives and limits.

losin
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Let f is monotone increasing, bounded, and differentiable on (a,inf)

Then does it necessarily follow that lim(f'(x),x,inf)=0 ?

It is hard to guess intuitively or imagine a counterexample...
 
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I will give you an 'intuitive' hint of what is going on. Since the function f is increasing and most importantly bounded, it means that as x-->infty, f(x) aproaches some point. IN other words |f(x)|<M, for some M, for all x. When you are taking the derivative of f, you are really talking about the slope of the tangent line at each point of f. since f is bounded by M, it means that as x->infty, f must get flater and flater. as a result the slopes of the tangent lines must also get smaller and smaller, eventually approaching zero.

In this case if A={f(x)|x in (a,infty)}, then f(x)-->sup{A} as x-->infty.
 
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