Graph Intersections: f'(x) > g'(x) - Can It Intersect More Than Once?

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Discussion Overview

The discussion centers around the question of whether two functions, f(x) and g(x), can intersect more than once given that their derivatives satisfy the condition f'(x) > g'(x) for all values of x. The scope includes mathematical reasoning and conceptual clarification related to the behavior of functions and their derivatives.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that the graphs can intersect only once.
  • Another participant argues that if the graphs were to intersect more than once, the slopes would not maintain the inequality f'(x) > g'(x).
  • A different perspective suggests that from a physics standpoint, if f'(x) > g'(x), then f(x) is always increasing at a faster rate than g(x), implying that after an intersection, g(x) cannot meet f(x) again unless g'(x) exceeds f'(x), which contradicts the initial condition.

Areas of Agreement / Disagreement

Participants generally agree that the graphs can intersect only once, but the reasoning behind this conclusion is debated, with different perspectives offered on the implications of the derivative condition.

Contextual Notes

The discussion does not resolve the mathematical implications of the derivative conditions fully, and assumptions about the behavior of the functions are not explicitly stated.

haxan7
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First of all i am new to this forum, and since my question is from my exam i did not post in homework section, if posted it in wrong section then kindly move it to right one.

So i had this mcq in exam it said that if f'(x)>g'(x) for all values of x.
then the graphs of two function intersect at.
(a) At only one point
(b)Many intersect more than once.

There were other options too but i did not mention them here because i know they were wrong.

I know that the graphs will intersect once, i want to know if it can intersect more than once.

Please also give your reasoning.
 
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The graphs can intersect only once.
 
because the slopes will not follow the similar inequality, if they are intersecting more than once.
 
You can think of it from a physics point of view. f'(x)>g'(x) means that rate of change of f(x) is always faster than g(x), once they intersect, the curve g(x) cannot meet f(x) unless until the rate of change of g(x) is greater than f(x) i.e. g'(x)>f'(x) which contradicts our original statement.
 
vivek.iitd said:
You can think of it from a physics point of view. f'(x)>g'(x) means that rate of change of f(x) is always faster than g(x), once they intersect, the curve g(x) cannot meet f(x) unless until the rate of change of g(x) is greater than f(x) i.e. g'(x)>f'(x) which contradicts our original statement.
Yeah got it, thanks
 

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