univmathstud
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- May 4, 2012
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Suppose f(x) and g(x) are two polynomials with non-negative coeficients. Further assume \(\displaystyle 0=f(0) \geq g(0)\) and \(\displaystyle f(1)=g(1)=1\). Show that \(\displaystyle (g'(x)-f'(x))(1-x) + g(x) - f(x) \geq 0\) on [0,1].
My idea so far:
f and g are convex functions. Therefore, \(\displaystyle f(x)+(1-x)f'(x) \leq 1\). This doesn't seem enough though.
Also, because f(1)=g(1)=1, the coefficients of the polynomials sum to one. But I have no idea how to use that.
Any idea on how to attack this problem?
My idea so far:
f and g are convex functions. Therefore, \(\displaystyle f(x)+(1-x)f'(x) \leq 1\). This doesn't seem enough though.
Also, because f(1)=g(1)=1, the coefficients of the polynomials sum to one. But I have no idea how to use that.
Any idea on how to attack this problem?
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