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 |a_1sinx+a_2sin2x++a_nsinn x|lt=|sinx| for x in R , then prove that |a_1+2a_1+3a+3+n a_n|lt=1

Show that 1+x ln (x+sqrt(x^2+1))geqsqrt(1+x^2) for all xgeq0.

If (1+x+x^2++x^p)^n=a_0+a_1x+a_2x^2++a_(n p)x^(n p), then find the value of a_1+2a_2+3a_3+ddot+n pa_(n p)dot

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

Let fa n dg be differentiable on R and suppose f(0)=g(0)a n df^(prime)(x)lt=g^(prime)(x) for all xgeq0. Then show that f(x)lt=g(x) for all xgeq0.

Let f(x)=a_0+a_1x+a_2x^2++a_n x^n+ and (f(x))/(1-x)=b_0+b_1x+b_2x^2++b_n x^n+ , then b_n+b_(n-1)=a_n b. b_n-b_(n-1)=a_n c. b_n//b_(n-1)=a_n d. none of these

If sum_(r=0)^n{a_r(x-alpha+2)^r-b_r(alpha-x-1)^r}=0, then prove that b_n-(-1)^n a_n=0.

Let P(x)=a_0+a_1x^2+a_2x^4++a_n x^(2n) be a polynomial in a real variable x with 0

Let n be a positive integer and (1+x+x^2)^n=a_0+a_1x++a^(2n)x^(2n)dot Show that a_0^2-a_1^ 2+a_2^ 2++'a_2n' x^2=a_ndot