KCET2010MathematicsContinuity and Differentiability
The function f(x)=|x-2|+x is
Options
- Adifferentiable at both x=2 and x=0
- Bdifferentiable at x=2 but not at x=0
- Ccontinuous at x =2 but not at x =0
- Dcontinuous at both x =2 and x =0
Correct answer
D. continuous at both x =2 and x =0
Step-by-step solution
f(x)=|x-2|+x First we check the continuity At x =0 , aligned RHL f(0+h) &= _ h 0 |0+h-2|+(0+h) &=|-2|=2 LHL f(0-h) &= _ h 0 |0-h-2|+(0-h) &=|-2|=2 aligned At x =2 , aligned RHL f (2+ h ) &= _ h 0 |2+ h -2|+(2+ h ) &=0+2+0=2 aligned LHL f(2-h)= _ h 0 |2-h-2|+(2+h) =0+2-0=2 and f(0)=2, f(2)=2 Hence, f(x) , is continuous at x=0,2 Now, we check differentiability f(x)= array c (-x+2-x), x < 0 (-x+2+x), 0 x 2 (x-2+2), 2 x array . f(x)= array c 2-2 x, x < 0 2,0 x 2 2 x-2,2 x array . Now, f^ (x)= array c -2, x < 0 0,0 x 2