Question

The function f(x) is defined as |[x]x| for −1<x≤2. The number of points where f(x) is non differentiable is

Solution

f=|[x]x|=|[x]| |x|

f(x)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩−x,−1<x<00,0≤x<1x,1≤x<24,x=2

L.H.L=limx→1−f(x)=0

R.H.L=limx→1+f(x)=1

L.H.L≠R.H.L

L.H.L=limx→2−f(x)=2

R.H.L=limx→2+f(x)=4

L.H.L≠R.H.L

f(x) discontinuous at x=1,2

Clearly, f(x) is not differentiable at x=1,2 and also f(x) is not differentiable at x=0

Hence, the number of points where f(x) is non differentiable is 3.

f(x)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩−x,−1<x<00,0≤x<1x,1≤x<24,x=2

L.H.L=limx→1−f(x)=0

R.H.L=limx→1+f(x)=1

L.H.L≠R.H.L

L.H.L=limx→2−f(x)=2

R.H.L=limx→2+f(x)=4

L.H.L≠R.H.L

f(x) discontinuous at x=1,2

Clearly, f(x) is not differentiable at x=1,2 and also f(x) is not differentiable at x=0

Hence, the number of points where f(x) is non differentiable is 3.

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