If the function f (x)= array ccc x+a 2 x & if & 0 x 4 2 x x+ b & if & 4 < x 2 a 2 x- bsin x & if & 2 < x array . is continuous in [0, ] then a-b=

Step-by-step solution

Continuity requires the function to match at transition points x = 4 and x = 2 . At x = 4 , the left limit equals the function value: _ x 4 ^- f(x) = f ( 4 ) = 4 + a . The right limit is _ x 4 ^+ f(x) = 2 + b . Equating gives 4 + a = 2 + b , yielding a - b = 4 . At x = 2 , the function value is f ( 2 ) = b . The right limit is _ x 2 ^+ f(x) = -a - b . Equating gives b = -a - b , or a + 2b = 0 . Solving the system a - b = 4 and a + 2b = 0 : From the second equation, a = -2b . Substituting into the first: -2b - b = 4