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)dot` Also, `f^(prime)(x)` is strictly increasing in `(a , c)dot` Prove that `(b-c)f(a)+(c-a)f(b)+(a-b)f(c)<0.`

If a , b , c , are three unequal number such that a , b ,c are in A.P and (b-a),(c-b) , a are in G.P then find a: b: c .

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, c be three number such that a+2b+4c=0. Then the equation ax^2+bx+c=0

Let a, b, c be real numbers such that a+b+clt0 and the quadratic equation ax^(2)+bx+c=0 has imaginary roots. Then

Let a and b be two real numbers such that a > 1, b >1. If A=((a,0),(0,b)), then (lim)_(n->oo) A^-n is (a) unit matrix (b) null matrix (c) 2I (d) non of these

Let a, b , c, p , q , r be positive real numbers such that a , b , c are in G.P. and a^p = b^q = c^r . Then

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 three unequal numbers such that a,b,c are in A.P. and (b-a) ,. (c-b) ,a are in G.P., then the value of a:b: c is-

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] If the point P(a,b,c) with reference to (E), lies on the plane 2x+y+z=1 , the the value of 7a+b+c is

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