Let `a , b , c` be three real numbers such that `a < b < c . f(x)` is continuous in `[a , c]` and differentiable in `(a , c)` Also, `f^(prime)(x)` is strictly increasing in `(a , c)` Prove that `(b-c)f(a)+(c-a)f(b)+(a-b)f(c)<0.`

Let a ,\ b ,\ c be three non-zero real numbers such that the equation sqrt(3)\ acosx+2\ bsinx=c ,\ x in [-pi/2,pi/2] , has two distinct real roots alpha and beta with alpha+beta=pi/3 . Then, the value of b/a is _______.

Let a ,b and c be real numbers such that a+2b+c=4 . Find the maximum value of (a b+b c+c a)dot

Let a ,b ,and c be real numbers such that 4a+2b+c=0 and a b > 0. Then the equation a x^2+b x+c = 0

Let a, b and c be real numbers such that 4a + 2b + c = 0 and ab gt 0. Then the equation ax^(2) + bx + c = 0 has

Let a, b and c be real numbers such that a - 7b + 8c = 4 and 8a + 4b - c = 7. What is the value of a^(2) - b^(2) + c^(2) .

Let a, b and c be three real numbers satisfying [a" "b" "c"][{:(1,9,7),(8,2,7),(7,9,7):}]=[0"" "0""" "0] and alpha and beta be the roots of the equation ax^(2) + bx + c = 0 , then sum_(n=0)^(oo) ((1)/(alpha)+(1)/(beta))^(n) , is

Let a,b and c be real numbers such that 4a+2b+c=0 and ab gt 0 .Then the equation ax^2+bx+c=0 has (A) real roots (B) Imaginary roots (C) exactly one root (D) roots of same sign

If a, b, c are positive real numbers such that a + b + c = 1 , then prove that a/(b + c)+b/(c+a) + c/(a+b) >= 3/2

Let a,b, and c be three real numbers satistying [a,b,c][(1,9,7),(8,2,7),(7,3,7)]=[0,0,0] Let omega be a solution of x^3-1=0 with Im(omega)gt0. I fa=2 with b nd c satisfying (E) then the vlaue of 3/omega^a+1/omega^b+3/omega^c is equa to (A) -2 (B) 2 (C) 3 (D) -3

Let a,b, and c be three real numbers satistying [a,b,c][(1,9,7),(8,2,7),(7,3,7)]=[0,0,0] Let b=6, with a and c satisfying (E). If alpha and beta are the roots of the quadratic equation ax^2+bx+c=0 then sum_(n=0)^oo (1/alpha+1/beta)^n is (A) 6 (B) 7 (C) 6/7 (D) oo