Define f(x)= array cc x^2+b x+c, & x < 1 x, & x 1 array . . If f(x) is differentiable at x=1 , then (b-c) is equal to

Step-by-step solution

Given function, aligned & f(x)= array cc x^2+b x+c, & x < 1 x, & x 1 array . & f^ (x)= array cc 2 x+b, & x < 1 1, & x 1 array . aligned Since, f(x) is differentiable at x=1 . aligned & _ x 1⁺ f^ (x)= _ x 1⁺ f^ (x) & _ x 1⁺ (2 x+b)= _ x 1⁺ 1 & 2+b=1 b=-1 aligned As form is differentiable at n=1 So, it will be continuous at x=1 also. aligned & _ x 1⁺ f(x)= _ x 1⁺ f^ (x)=f(1) & _ x 1⁺ x^2+b x+c= _ x 1⁺ x=1 & 1+b+c=1 & 1-1+c=1 c=1 aligned Hence, b-c=-1-1=-2