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

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.

If f(x) = (a-x^n)^(1/n), a > 0 and n in N , then prove that f(f(x)) = x for all x.

Let u(x) and v(x) satisfy the differential equation (d u)/(dx)+p(x)u=f(x) and (d v)/(dx)+p(x)v=g(x) are continuous functions. If u(x_1) for some x_1 and f(x)>g(x) for all x > x_1, prove that any point (x , y), where x > x_1, does not satisfy the equations y=u(x) and y=v(x)dot

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

If phi (x) is a differentiable real valued function satisfying phi (x) +2phi le 1 , then it can be adjucted as e^(2x)phi(x)+2e^(2x)phi(x)lee^(2x) or (d)/(dx)(e^(2)phi(x)-(e^(2x))/(2))le or (d)/(dx)e^(2x)(phi(x)-(1)/(2))le0 Here e^(2x) is called integrating factor which helps in creating single differential coefficeint as shown above. Answer the following question: If p(1)=0 and dP(x)/(dx)ltP(x) for all xge1 then

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)]

Suppose p(x)=a_0+a_1x+a_2x^2+...+a_n x^ndot If |p(x)|lt=|e^(x-1)-1| for all xgeq0, prove that |a_1+2a_2+...+n a_n|lt=1.

If f(x) defined by f(x)={(|x^2-x|)/(x^2-|x|),x!=0,1-1 . Then (A)f(x) is continuous for all x (B) for all x except x=0 (C) for all x except x=1 (D) for all x except x=0 and x=1

If f(x) defined by f(x)={(|x^2-x|)/(x^2-|x|),x!=0,1-1 . Then (A)f(x) is continuous for all x (B) for all x except x=0 (C) for all x except x=1 (D) for all x except x=0 and x=1