If `f(x)=|x+a^2a b a c a b x+b^2b c a c b c x+c^2|,t h e n` prove that `f^(prime)(x)=3x^2+2x(a^2+b^2+c^2)dot`

If a , b , c are real numbers, then find the intervals in which f(x)=|x+a^2a b a c a b x+b^2b c a c b c x+c^2| is increasing or decreasing.

The determinant |y^2-x y x^2a b c a ' b ' c '| is equal to a. |b x+a y c x+b y b^(prime)x+a ' y c^(prime)x+b ' y| b. |a x+b y b x+c y a^(prime)x+b ' y b ' x+c ' y| c. |b x+c y a x+b y b^(prime)x+c ' y a^(prime)x+b ' y| d. |a x+b y b x+c y a^(prime)x+b ' y b^(prime)x+c ' y|

If f(x)=a x^2+b x+a^2+b^2+c^2-a b-b c-c a ,w h e r ea ,b ,c are distinct reals, has imaginary roots then (a) 2(a-b)+(a-b)^2+(b-c)^2+(c-a)^2>0 (b) f(0)>0 (c) f(-1)>0 (d)f(0)<0

If roots of equation a-b)x^(2)+(b-c)x+(c-a)=0 are equal then prove that b+c=2a

(x^(a^2 + b^2)/x^(ab))^(a+b) (x^(b^2 + c^2)/x^(bc))^(b+c) (x^(c^2 + a^2)/x^(ac))^(a+c) = x^(2(a^3+b^3+c^3))

Suppose A, B, C are defined as A=a^2b+a b^2-a^2c-a c^2, B=b^2c+b c^2-a^2b-a b^2, a n dC=a^2c+a c^2-b^2c-b c^2, w h e r ea > b > c >0 and the equation A x^2+B x+C=0 has equal roots, then a ,b ,c are in AdotPdot b. GdotPdot c. HdotPdot d. AdotGdotPdot

Let A=[[3x^2], [1], [6x]],B=[a,b,c]and C=[[(x+2)^2, 5x^2, 2x],[5x^2, 2x,(x+2)^2],[ 2x,(x+2)^2, 5x^2]] be three given matrices, where a ,b ,ca n dx in Rdot Given that tr(AB)=tr(C). If f(x)=a x^2+b x+c , then the value of f(1) is ______.