Q: Let f be the function defined by f(x)= \ array cc x^2-1 x^2-2|x-1|-1 , & x 1 \\ 1 2 , & x=1 array .

f(x)= \ array cc x^2-1 x^2-2 x+1 ; & x 1 \\ 1 2 ; & x=1 \\ x^2-1 x^2+2 x-3 ; & x 1 array . x^2+3 x-x-3 =x(x+3)-1(x+3) Lets Find LHL, _ x 1^ - x^2-1 x^2-2 x+1 and RHL, _ x 1^ + x^2-1 x^2+2 x-3 aligned & _ x 1^ - (x-1)(x+1) (x-1)^2 _ x 1^ + (x-1)(x+1) (x+3)(x-1) \\ & _ x 1^ - x+1 x-1 = _ x 1^ + x+1 x+3 = 2 4 = 1 2 aligned LHL f(1) f(x) is not continuous at x=1

Question 1

Let f be the function defined by f(x)= \ array cc x^2-1 x^2-2|x-1|-1 , & x 1 \ 1 2 , & x=1 array .

Accepted Answer

f(x)= \ array cc x^2-1 x^2-2 x+1 ; & x 1 \ 1 2 ; & x=1 \ x^2-1 x^2+2 x-3 ; & x 1 array . x^2+3 x-x-3 =x(x+3)-1(x+3) Lets Find LHL, _ x 1^ - x^2-1 x^2-2 x+1 and RHL, _ x 1^ + x^2-1 x^2+2 x-3 aligned & _ x 1^ - (x-1)(x+1) (x-1)^2 _ x 1^ + (x-1)(x+1) (x+3)(x-1) \ & _ x 1^ - x+1 x-1 = _ x 1^ + x+1 x+3 = 2 4 = 1 2 aligned LHL f(1) f(x) is not continuous at x=1

Question 2

If f(x)= \ array cc x^2 ( x) (1+x) , & x 0 \ 0, & x=0 array ., then at x=0, f(x) is

Accepted Answer

_ x 0 f(x)= _ x 0 x^2 ( x) (1+x) = _ x 0 - x ( x) (1+x) x = _ x 0 x ( x)=0 1=0 _ x 0 f(x)=f(0) f(x) is continuous at x=0 Now, _ x 0 f(x+h)-f(x) h _ h 0 f(h)-f(0) h = _ h 0 h^2 h-0 (1+h) _ h 0 h^2 ( h) 1 =0 f(x) is differentiable at x=0

Question 3

f(x)= \ array cc 4 & - x - 5 \ x^2-1 & - 5 x 5 \ 4 & 5 x array . If k is the number of points where f(x) is not differentiable then k-2=

Accepted Answer

Here f(x) is continous x R Now At x=- 5 L.H.D = 0 R.H.D =2 x=-2 5 f ( x ) is not differentiable at x =- 5 At x= 5 L.H.D. =2 x=2 5 R.H.D. =0 f ( x ) is not differentiable at x = 5 k =2 k-2=2-2=0

Question 4

Let f be the function defined by f(x)= \ array ll x^ 2 -1 x^ 2 -2|x-1|-1 , & x 1 \ 1 / 2, & x=1 array .

Accepted Answer

For x 1, f(x)= x^ 2 -1 x^ 2 -2 x+1 = x+1 x-1 _ x 1^ + f(x)= The function is not continuous at x =1.

Continuity and Differentiability — Previous Year Questions & Solutions