Paper 1, Section II, E

Analysis I | Part IA, 2016

Let f:RRf: \mathbb{R} \rightarrow \mathbb{R}. We say that xRx \in \mathbb{R} is a real root of ff if f(x)=0f(x)=0. Show that if ff is differentiable and has kk distinct real roots, then ff^{\prime} has at least k1k-1 real roots. [Rolle's theorem may be used, provided you state it clearly.]

Let p(x)=i=1naixdip(x)=\sum_{i=1}^{n} a_{i} x^{d_{i}} be a polynomial in xx, where all ai0a_{i} \neq 0 and di+1>did_{i+1}>d_{i}. (In other words, the aia_{i} are the nonzero coefficients of the polynomial, arranged in order of increasing power of xx.) The number of sign changes in the coefficients of pp is the number of ii for which aiai+1<0a_{i} a_{i+1}<0. For example, the polynomial x5x3x2+1x^{5}-x^{3}-x^{2}+1 has 2 sign changes. Show by induction on nn that the number of positive real roots of pp is less than or equal to the number of sign changes in its coefficients.

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