Let f_ k (x)= 1 k ( ^ k x+ ^ k x ) for k =1,2,3, Then for all x R , the value of f₄(x)-f₆(x) is equal to :

Step-by-step solution

f_ k (x)= 1 k ( ^ k x+ ^ k x )f₄(x)= 1 4 [ ⁴ x+ ⁴ x ]= 1 4 [ ( ² x+ ² x )²- ( 2 x)² 2 ]= 1 4 [1- ( 2 x)² 2 ]f₆(x)= 1 6 [ ⁶ x+ ⁶ x ]= 1 6 [ ( ² x+ ( ² x )- 3 4 ( ² x )² ] .= 1 6 [1- 3 4 ( 2 x)² ] Now f₄(x)-f_ (6) (x)= 1 4 - 1 6 - ( 2 x)² 8 + 1 8 ( 2 x)²= 1 12