JEE MAIN - Mathematics (2022 - 28th June Morning Shift - No. 6)
Let f : R $$\to$$ R be defined as
$$f(x) = \left[ {\matrix{ {[{e^x}],} & {x < 0} \cr {a{e^x} + [x - 1],} & {0 \le x < 1} \cr {b + [\sin (\pi x)],} & {1 \le x < 2} \cr {[{e^{ - x}}] - c,} & {x \ge 2} \cr } } \right.$$
where a, b, c $$\in$$ R and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?
Explanation
$$f(x) = \left\{ {\matrix{ 0 & {x < 0} \cr {a{e^x} - 1} & {0 \le x < 1} \cr b & {x = 1} \cr {b - 1} & {1 < x < 2} \cr { - c} & {x \ge 2} \cr } } \right.$$
To be continuous at x = 0
a $$-$$ 1 = 0
to be continuous at x = 1
ae $$-$$ 1 = b = b $$-$$ 1 $$\Rightarrow$$ not possible
to be continuous at x = 2
b $$-$$ 1 = $$-$$ c $$\Rightarrow$$ b + c = 1
If a = 1 and b + c = 1 then f(x) is discontinuous at exactly one point.
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