Let f(x)= [ array l x^2+a, 0 x 1 2 x+b, 1 x 2 array . and g(x)= [ array ll 3 x+b, & 0 x 1 x^3 & , 1 x 2 array . If derivative of f(x) w.r.t. g(x) at x=1 exists, and has value equal to m then the value of |a+b+6 m| is equal to

Step-by-step solution

f ^ (1⁺ )= Lim _ h 0 2(1+h)+b-(2+b) h =2f^ (1⁻ )= Lim _ h 0 (1-h)^2+a-(2+b) -h = Lim _ h 0 (h^2-2 h )+a-b-1 -h =2 where a-b-1=0 (1)g^ (1⁻ )= Lim _ h 0 3(1-h)+b-1 -h =3 where 2+b=0 b=-2 (2)g^ (1⁺ )= Lim _ h 0 (1+h)^3-1 h = Lim _ h 0 1+3 h+3 h^2+h^3-1 h =3 from (1) and (2) a = -1; f ^ ( x ) g ^ ( x ) at x =1= 2 3