Monday, 4 February 2013

real analysis - Minimum difference of roots of a polynomial and its derivative

Let P(x)=(xx1)(xx2)...(xxn) where all the n roots are real and distinct. Let y1,y2,...,yn1 be the roots of P. Show that



min.



My thoughts: We have P'(x) = P(x)(\frac1{x-x_1}+...+\frac1{x-x_n}). So we may consider P'(x)/P(x), which has poles at the roots of P.

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real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

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