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 P(1)=0a n d(d P(x))/(dx)>P(x) for all x>=1. Prove that P(x)>0 for all x>1

If P(1)=0and(dP(x))/(dx)>P(x), for all x>=1 Prove that P(x)>0 for all x>1

Let a, b, c in R with a gt 0 such that the equation ax^(2) + bcx + b^(3) + c^(3) - 4abc = 0 has non-real roots. If P(x) = ax^(2) + bx + c and Q(x) = ax^(2) + cx + b , then (a) P(x) gt 0 for all x in R and Q(x) lt 0 for all x in R . (b) P(x) lt 0 for all x in R and Q(x) gt 0 for all x in R . (c) neither P(x) gt 0 for all x in R nor Q(x) gt 0 for all x in R . (d) exactly one of P(x) or Q(x) is positive for all real x.

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

If x gt 0 , prove that, x gt log (1+x) gt x/(1+x)

(d)/(dx) (x^(x))= ….(x gt 0)

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)]

f(x)=e^(-1/x),w h e r ex >0, Let for each positive integer n ,P_n be the polynomial such that (d^nf(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)]