2011-4 A polynomial with distinct real zeros

Let n>2. Let f (x) be a degree-n polynomial with real coefficients. If f (x) has n distinct real zeros r1<r2<…<rn, then Rolle’s theorem implies that the largest real zero q of (x) is between rn-1 and rn. Prove that q>(rn-1+rn)/2.

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2011-4 A polynomial with distinct real zeros, 2.6 out of 5 based on 14 ratings