If the function f: R R defined by f(x)= array cc (a+1) x+ x x , & x < 0 x+x^2 - x x^ 3 / 2 , & x=0 is array . continuous on [R , then a+b=

Step-by-step solution

Given function f(x)= array cc (a+1) x+ x x & , x 0 array . Function is continuous so, _ x 0⁻ f(x)= _ x 0⁺ f(x)=f(0) Now, aligned & _ x 0⁻ f(x)= _ x 0⁻ ( (a+1) x+ x x ) = & _ x 0⁻ ( (a+1) x (a+1) (a+1) x + x x ) = & (a+1)+1 & [ _ x 0⁻ x x =1 ] aligned Now, aligned _ x 0⁺ f(x) & = _ x 0⁺ + x+x^2 - x x^ 3 / 2 = & _ x 0⁺ x ( 1+x -1) 1+x +1 x x ( 1+x +1) aligned b= 1 2 put in Eq. (iii) aligned a+2 & = 1 2 a & = -3 2 aligned Now, aligned & a+b= -3 2 + 1 2 & a+b=-1 aligned