If f: R R defined by f(x)= array cc 1+3 x^2- 2 x x^2 , & for x 0 & , for x=0 array . is continuous at x=0 , then k is equal to

Step-by-step solution

f(x)= array rr 1+3 x^2- 2 x x^2 , & for x 0 k, & for x=0 array . RHL f(0+h)= _ h 0 1+3(0+h)^2- 2(0+h) (0+h)^2 = _ h 0 1+3 h^2- 2 h h^2 = _ h 0 1+3 h^2- (1-2 ^2 h ) h^2 = _ h 0 1+3 h^2-1+2 ^2 h h^2 = _ h 0 3+2 ( ^2 h h^2 ) =3+2 _ h 0 ( h h )^2=3+2 (1)^2 _ x 0 x x =1 =3+2=5 LHL f(0-h)= _ h 0 1+3(0-h)^2- 2(0-h) (0-h)^2 = _ h 0 1+3 h^2- 2 h h^2 =5 Since, the function is continuous at x=0 , then LHL = RHL =f(0) f(0)=k k=5