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

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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PGDBA 2019

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

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