If `P(1)=0a n d(d P(x))/(dx)gtP(x)`, for all ` xge1`. Prove that `P(x)>0` for all `x>1.`

If f'(x) gt f(x)" for all "x ge 1 and f(1)=0 , then

Prove that tan x gt x for all x in [0, (pi)/(2)]

If H(x_(0)) =0 for some x= x_(0) and (d)/(dx)H(x)gt2cxH(x) for all xgex_(0) where cgt0 then

Let P(x) be a polynomial such that P(1)=1 and (p(2x))/(p(x+1))=8-(56)/(x+7) for all real x for which both sides are defined.The degree of P(x) is

f(x)=e^(-(1)/(x)), where x>0, Let for each positive integer n,P_(n) be the polynomial such that (d^(n)f(x))/(dx^(n))=P_(n)((1)/(x))e^(-(1)/(x)) for all x>0. show that P_(n+1)(x)=x^(2)[P_(n)(x)-(d)/(dx)P_(n)(x)]

A polynomial P(x) with real coefficients has the property that P^(n)(x)ne0 for all x. Suppose P(0)=1 and P'(0) = -1 . What can you say about P(1)?

Let P be a non-zero polynomial such that P(1+x)=P(1-x) for all real x, and P(1)=0. Let m be the largest integer such that (x-1)^m divides P(x) for all such P(x). Then m equals