Let `(a_0)/(n+1)+(a_1)/n+(a_2)/(n-1)++(a_(n-1))/2+a_n=0.` Show that there exists at least real `x` between 0 and 1 such that `a_0x^n+a_1x^(n-1)+a_2x^(n-2)++a_n=0`

Differentiate the following with respect of x : a_0x^n+a_1x^(n-1)+a_2x^(n-2)++a_(n-1)x+a_ndot

If a_1, a_2, a_3 ......a_n (n>= 2) are real and (n-1) a_1^2 -2na_2 < 0 then prove that at least two roots of the equation x^n+a_1 x^(n-1) +a_2 x^(n-2) +......+a_n = 0 are imaginary.

If the polynomial equation a_n x^n + a_(n-1) x^(n-1) + a_(n-2) x^(n-2) + ....... + a_0 = 0, n being a positive integer, has two different real roots a and b . then between a and b the equation na_n x^(n-1) +(n-1)a_(n-1) x^(n-2) +.......+a_1 =0 has

Write the first five terms in each of the following sequence: a_1=1=\ a_2,\ a_n=a_(n-1)+a_(n-2),\ n >2

Let a_0x^n + a_1 x^(n-1) + ... + a_(n-1) x + a_n = 0 be the nth degree equation with a_0, a_1, ... a_n integers. If p/q is arational root of this equation, then p is a divisor of an and q is a divisor of a_n . If a_0 = 1 , then every rationalroot of this equation must be an integer.

If each a_i > 0 , then the shortest distance between the points (0,-3) and the curve y=1+a_1x^2+a_2 x^4+.......+a_n x^(2n) is

Write the first five terms in each of the following sequence: a_1=a_2=2,\ a_n=a_(n-1)-1,\ n >1 .

If a_1,a_2 …. a_n are positive real numbers whose product is a fixed real number c, then the minimum value of 1+a_1 +a_2 +….. + a_(n-1) + a_n is :

Given a_1=1/2(a_0+A/(a_0)), a_2=1/2(a_1+A/(a_1)) and a_(n+1)=1/2(a_n+A/(a_n)) for n>=2 , where a>0,A>0 . prove that (a_n-sqrt(A))/(a_n+sqrt(A))=((a_1-sqrt(A))/(a_1+sqrt(A)))2^(n-1) .

If d/(dx){x^n-a_1x^(n-1)+a_2x^(n-2)++(-1)^n a_n}e^x=x^n e^x , then the value of ar, 0 < r ≤ n, is equal to