Le^x-(1+x/1+x^2/2)l <= e/6 for all x E [0,1].

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Discussion Overview

The discussion revolves around demonstrating the inequality \( |e^x - (1 + \frac{x}{1!} + \frac{x^2}{2!})| \leq \frac{e}{6} \) for all \( x \) in the interval \([0,1]\). Participants explore methods to prove this, including boundary evaluations and monotonicity of the function involved.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant suggests verifying the inequality at the boundaries of the interval and checking if the left-hand side is monotonic in \( x \).
  • Another participant hints at using Taylor's theorem as a potential approach to the problem.
  • There is a discussion on how to show that a function is monotonic increasing, with one participant proposing a method that is later challenged as insufficient.
  • A clarification is provided regarding the definition of a monotonic increasing function, emphasizing that it requires \( f(x) \geq f(y) \) for \( x \geq y \).

Areas of Agreement / Disagreement

Participants have not reached a consensus on the best approach to prove the inequality. There are multiple competing views on how to establish monotonicity and the application of Taylor's theorem.

Contextual Notes

Some assumptions about the behavior of the function and the application of mathematical theorems are not fully explored, leaving certain steps unresolved.

squenshl
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I have a problem.
How do I show that le^x-(1+x/1!+x^2/2!)l <= e/6 for all x E [0,1].
 
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Show that the equation is true at the boundries. Show that the term on the left is monotonic in x (if, indeed, it is).
 


What a strange coincidence. I saw this thread only after posting my example in this thread.

If you don't want to see the full solution, here's a hint: Taylor's theorem.
 


how to show that a function is monotonic increasing? show that f(x+1) >= f(x) for all x?
 


ice109 said:
how to show that a function is monotonic increasing? show that f(x+1) >= f(x) for all x?

No, that won't work. A function is monotonic increasing if f(x) ≥ f(y) whenever x ≥ y.
 

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