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Question

If f (x) = |3 − x| + (3 + x), where (x) denotes the least integer greater than or equal to x, then f (x) is
(a) continuous and differentiable at x = 3
(b) continuous but not differentiable at x = 3
(c) differentiable nut not continuous at x = 3
(d) neither differentiable nor continuous at x = 3

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Solution

(d) neither differentiable nor continuous at x = 3

We have,fx=3-x+3+x, where x denotes the least integer greater than or equal to x.fx=3-x+3+3, 2<x<3-3+x+3+4, 3<x<4 fx=-x+9 2<x<3x+4 3<x<4Here,LHL at x=3=limx3-fx=limx3--x+9=-3+9=6RHL at x=3=limx3+fx=limx3-x+4=3+4=7Since, LHL at x=3RHL at x=3Hence, given function is not continuous at x= 3Therefore, the function will also not be differentiable at x=3

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