Suppose f(x)= [ array cc x^3 & if x 1 a x^2+b x+c & if x 1 array . . If f^ (1) exist then find the value of a^3+b^3+c^3 .

Step-by-step solution

For continuity at x=1 we leave f (1⁻ )=1 and f (1⁺ )=a+b+c a+b+c=1...(1)f^ (x)= [ array ll 3 x^2 & if x 1 2 a x+b & if x 1 array . for continuity of f ^ ( x ) at x =1f^ (1⁻ )=3 ; f^ (1⁺ )=2 a+b hence 2 a+b=3....(2) aligned & f^ (x)= [ array cc 6 x & if x 1 2 a & if x 1 array . & f^ (1⁻ )=6 ; f^ (1⁺ )=2 a & 2 a=6 & a=3 aligned from (2), b = - 3 from (2), c = 1 Hence a=3, b=-3 ; c=1 a ^3=1