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Suppose $a,b$ belongs to $\mathbb{R}$ and $a<b$.Suppose the functions $f$ and $g$ are continuous on $[a,b]$ and differentiable on $(a,b)$. suppose $f(a)>g(a)$, $f(b)<g(b)$,$g(x)>o$,where $x$ belongs to $[a,b]$ and $f'(x)>0$ ,$x$ belongs to $(a,b)$.

a. Show that there is $c$ belongs to $(a,b)$ such that 1. $g(c)f'(c)<f(c)g'(c)$ 2. $c$ is not a critical number of $g$

b. Show that there is $d$ belongs to $(a,b)$ such that for any $n$ belongs to $\mathbb{N}$ (Natural Numbers) , $(g'')(d)(f'')'(d)<(f'')(d)(g'')'(d)$

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What did you try? Where are you stuck? –  Davide Giraudo Apr 1 '12 at 12:00
Where does $n$ come into play in condition b.? –  David Mitra Apr 1 '12 at 14:54
-1: This question does not show any effort on the part of the asker. –  Rahul May 31 '12 at 22:23

1 Answer 1

Hint: let $c=\min\{x\in (a,b):f(x)=g(x)\}$, use continuity and the other givens to prove that such a value is well-defined and conclude the first part of a, which gives the second part immediately.

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