Let a, b (a 0) . If the function f is defined as f(x)= array cc 2 x^2 a & , 0 x 1 a & , 1 x 2 2 ~b ^2-4 ~b x & , 2 x array . is continuous in the interval [0, ) , then an ordered pair ( a , b ) is

Step-by-step solution

Function: f(x)= cases 2 x^2 a , & 0 x 1 a, & 1 x 2 2 b^2-4 b x , & 2 x cases Condition: Function f(x) must be continuous in the interval [0, ) . This implies: _ x 1⁻ f(x)= _ x 1⁺ f(x), and _ x 2 ⁻ f(x)= _ x 2 ⁺ f(x) . Continuity at x=1 : _ x 1⁻ f(x)= 2(1)^2 a = 2 a , _ x 1⁺ f(x)=a . Equating these: 2 a =a a^2=2 a= 2 Continuity at x= 2 : _ x 2 ⁻ f(x)=a, _ x 2 ⁺ f(x)= 2 b^2-4 b 2 Equating these: a= 2 b^2-4 b 2 a 2 =2 b^2-4 b . Substitute a= 2 : ( 2 )( 2 )=2 b^2-4 b 2=2 b^2-4 b Simplify: b^2-2 b-1=0 Solve for b : b= 2