Derivative of monotone increasing and bounded f

In summary, if a function f is monotone increasing, bounded, and differentiable on (a,inf), it can be deduced that the limit of f'(x) as x approaches infinity is equal to 0. This is because as x approaches infinity, the function f approaches a finite value and the slopes of the tangent lines must gradually decrease, leading to the limit of f'(x) approaching 0.
  • #1
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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  • #2
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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1. What is the definition of a monotone increasing function?

A monotone increasing function is a function where the output (or y-value) increases as the input (or x-value) increases. In other words, the function is constantly moving upwards and never decreases.

2. How do you find the derivative of a monotone increasing function?

To find the derivative of a monotone increasing function, you can use the power rule or the chain rule, depending on the specific function. The derivative will always be a positive value, as the function is constantly increasing.

3. What is the importance of a bounded function?

A bounded function is important because it means that the function's values are limited and do not continue to increase or decrease without bound. This allows us to make predictions and analyze the function more easily.

4. Can a function be monotone increasing and bounded at the same time?

Yes, a function can be both monotone increasing and bounded simultaneously. For example, the function f(x) = x on the interval [0,10] is both monotone increasing and bounded between 0 and 10.

5. How does the derivative of a monotone increasing and bounded function relate to its graph?

The derivative of a monotone increasing and bounded function will always be positive, which means that the slope of the graph will always be positive. This results in a graph that is constantly increasing and never decreases, with a limited range of values.

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