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Question 22
If $$f'(x)$$ and $$g'(x)$$ exist for all $$x \in R$$, and if $$f'(x)>g'(x)$$ for all $$x \in R$$, then the curve $$y = f(x)$$ and $$y = g(x)$$ in the $$xy$$-plane
Solution
Let a(x)=f(x)-g(x).
Then a'(x) = f'(x)-g'(x)
Now a'(x) is an increasing function. This would mean that a(x) is an increasing function, and hence one-one function. Thus, there can be atmost 1 point of intersection between the two graphs.
The number of intersections can be 0 or 1 depending upon the function.
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