Let f : ℝ → ℝ be defined as f x = e x , x < 0 a e x + x - 1 , 0 ≤ x < 1 b + sin π x , 1 ≤ x < 2 e - x - c , x ≥ 2 where a , b , c ∈ ℝ and t denotes greatest integer less than or equal to t . Then, which of the following statements is true?

Given f x = e x , x < 0 a e x + x - 1 , 0 ≤ x < 1 b + sin π x , 1 ≤ x < 2 e - x - c , x ≥ 2 i.e. f x = 0 , x < 0 a e x - 1 , 0 ≤ x < 1 b , x = 1 b - 1 , 1 < x < 2 - c , x ≥ 2 Here, we know f x is discontinuous at x = 1 as b ≠ b - 1 For f x to be continuous at x = 0 ; a = 1 For f x to be continuous at x = 2 ; b + c = 1 Hence, a + b + c = 2

Mathematics - Continuity - Differentiability Question with Solution | TestHub

MathematicsContinuity - DifferentiabilityContinuity- MiscellaneousMedium2 minPYQ_2022

MathematicsMediumsingle choice

Let $f : ℝ \to ℝ$ be defined as $f (x) = [\begin{array}{l} [e^{x}], & x < 0 \\ a e^{x} + [x - 1], & 0 \leq x < 1 \\ b + [\sin (π x)], & 1 \leq x < 2 \\ [e^{- x}] - c, & x \geq 2 \end{array}$
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?

Options:

Answer:

Solution:

Given $f (x) = [\begin{array}{l} [e^{x}], & x < 0 \\ a e^{x} + [x - 1], & 0 \leq x < 1 \\ b + [\sin (π x)], & 1 \leq x < 2 \\ [e^{- x}] - c, & x \geq 2 \end{array}$

i.e. $f (x) = \{\begin{matrix} 0, & x < 0 \\ a e^{x} - 1, & 0 \leq x < 1 \\ b, & x = 1 \\ b - 1, & 1 < x < 2 \\ - c, & x \geq 2 \end{matrix}$

Here, we know $f (x)$ is discontinuous at $x = 1$ as $b \neq b - 1$

For $f (x)$ to be continuous at $x = 0; a = 1$

For $f (x)$ to be continuous at $x = 2; b + c = 1$

Hence, $a + b + c = 2$

Stream:JEESubject:MathematicsTopic:Continuity - DifferentiabilitySubtopic:Continuity- Miscellaneous

⏱ 2mℹ️ Source: PYQ_2022

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