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`

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 one real x between 0 and 1 such that a_(0)x^(n) + a_(1)x^(n-1) + …..+ a_(n)=0

If the equation a_0x^n +a_1x^(n-1) +....a_(n-1)x=0 has a positive root alpha prove that the equation na_0x^(n-1) +(n-1)a_1 x^(n-2) + ---+a_(n-1) = 0 also has a positive root smaller than alpha

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

Evaluate lim_(xrarrc) f(x) where f(x) = a_nx^n+a_(n-1)x^(n-1)+a_(n-2)x^(n-2)+.....+a_2x^2+a_1x^1+a_0

A_1 A_2 A_3............A_n is a polygon whose all sides are not equal, and angles too are not all equal. theta_1 is theangle between bar(A_i A_j +1) and bar(A_(i+1) A_(i +2)),1 leq i leq n-2.theta_(n-1) is the angle between bar(A_(n-1) A_n) and bar(A_n A_1), while theta_n, is the angle between bar(A_n A_1) and bar(A_1 A_2) then cos(sum_(i=1)^n theta_i) is equal to-

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 :

Let f(x) = a_(0)x^(n) + a_(1)x^(n-1) + a_(2) x^(n-2) + …. + a_(n-1)x + a_(n) , where a_(0), a_(1), a_(2),...., a_(n) are real numbers. If f(x) is divided by (ax - b), then the remainder is