JEE Advance - Mathematics (2020 - Paper 2 Offline - No. 5)
Let the functions $$f:( - 1,1) \to R$$ and $$g:( - 1,1) \to ( - 1,1)$$ be defined by $$f(x) = |2x - 1| + |2x + 1|$$ and $$g(x) = x - [x]$$, where [x] denotes the greatest integer less than or equal to x. Let $$f\,o\,g:( - 1,1) \to R$$ be the composite function defined by $$(f\,o\,g)(x) = f(g(x))$$. Suppose c is the number of points in the interval ($$-$$1, 1) at which $$f\,o\,g$$ is NOT continuous, and suppose d is the number of points in the interval ($$-$$1, 1) at which $$f\,o\,g$$ is NOT differentiable. Then the value of c + d is ............
Answer
4
Explanation
The given functions $$f:( - 1,1) \to R$$ and $$g:( - 1,1) \to ( - 1,1)$$ be defined by $$f(x) = |2x - 1| + |2x + 1|$$ and $$g(x) = x - [x]$$.
As, we know the composite function (fog) (x) is discontinuous at the points, where g(x) is discontinuous for given domain. And, since g(x) is discontinuous at x = 0 lies in interval ($$-$$1, 1), so value of c = 1.
And, since (fog) (x) is not differentiable at the point where g(x) is not differentiable as well as at those points also where g(x) attains the values so that f(g(x)) is non-differentiable.
Since g(x) is not continuous at x = 0$$ \in $$ ($$-$$1, 1) so fog(x) is not differentiable and as $$f(x) = |2x - 1| + |2x + 1|$$ is not differentiable at x = $$-$$1/2 and 1/2, so (fog) (x) is not differentiable for those x, for which g(x) = $$-$$1/2 or 1/2.
But g(x) $$ \ge $$ 0, so g(x) can be 1/2 only and for x = $$-$$1/2 and 1/2, g(x) = $${1 \over 2}$$.
So, (fog) (x) is not differentiable at x = $$-$$1/2, 0, 1/2, therefore value of d = 3
$$ \therefore $$ c + d = 1 + 3 = 4.
As, we know the composite function (fog) (x) is discontinuous at the points, where g(x) is discontinuous for given domain. And, since g(x) is discontinuous at x = 0 lies in interval ($$-$$1, 1), so value of c = 1.
And, since (fog) (x) is not differentiable at the point where g(x) is not differentiable as well as at those points also where g(x) attains the values so that f(g(x)) is non-differentiable.
Since g(x) is not continuous at x = 0$$ \in $$ ($$-$$1, 1) so fog(x) is not differentiable and as $$f(x) = |2x - 1| + |2x + 1|$$ is not differentiable at x = $$-$$1/2 and 1/2, so (fog) (x) is not differentiable for those x, for which g(x) = $$-$$1/2 or 1/2.
But g(x) $$ \ge $$ 0, so g(x) can be 1/2 only and for x = $$-$$1/2 and 1/2, g(x) = $${1 \over 2}$$.
So, (fog) (x) is not differentiable at x = $$-$$1/2, 0, 1/2, therefore value of d = 3
$$ \therefore $$ c + d = 1 + 3 = 4.
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