Class CBSE Class 12 Mathematics Continuity and Differentiability Q #599
COMPETENCY BASED
APPLY
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
If f(x)=∣x∣+∣x−1∣, then which of the following is correct?
(A) f(x) is both continuous and differentiable, at x=0 and x=1.
(B) f(x) is differentiable but not continuous, at x=0 and x=1.
(C) f(x) is continuous but not differentiable, at x=0 and x=1.
(D) f(x) is neither continuous nor differentiable, at x=0 and x=1.
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Correct Answer: C

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Detailed Solution

Step 1: Define the function f(x) piecewise

The function $f(x) = |x| + |x-1|$ can be defined piecewise as follows: If $x < 0$, then $|x| = -x$ and $|x-1| = -(x-1) = 1-x$. So, $f(x) = -x + 1 - x = 1 - 2x$. If $0 \le x < 1$, then $|x| = x$ and $|x-1| = -(x-1) = 1-x$. So, $f(x) = x + 1 - x = 1$. If $x \ge 1$, then $|x| = x$ and $|x-1| = x-1$. So, $f(x) = x + x - 1 = 2x - 1$. Thus, $$f(x) = \begin{cases} 1-2x, & x < 0 \\ 1, & 0 \le x < 1 \\ 2x-1, & x \ge 1 \end{cases}$$

Step 2: Check continuity at x=0

For continuity at $x=0$, we need to check if $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)$. $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (1-2x) = 1 - 2(0) = 1$. $\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (1) = 1$. $f(0) = 1$. Since $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0) = 1$, $f(x)$ is continuous at $x=0$.

Step 3: Check differentiability at x=0

For differentiability at $x=0$, we need to check if the left-hand derivative (LHD) equals the right-hand derivative (RHD). LHD at $x=0$: $f'(0^-) = \frac{d}{dx}(1-2x) = -2$. RHD at $x=0$: $f'(0^+) = \frac{d}{dx}(1) = 0$. Since $f'(0^-) \ne f'(0^+)$, $f(x)$ is not differentiable at $x=0$.

Step 4: Check continuity at x=1

For continuity at $x=1$, we need to check if $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1)$. $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (1) = 1$. $\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2x-1) = 2(1) - 1 = 1$. $f(1) = 2(1) - 1 = 1$. Since $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1) = 1$, $f(x)$ is continuous at $x=1$.

Step 5: Check differentiability at x=1

For differentiability at $x=1$, we need to check if the left-hand derivative (LHD) equals the right-hand derivative (RHD). LHD at $x=1$: $f'(1^-) = \frac{d}{dx}(1) = 0$. RHD at $x=1$: $f'(1^+) = \frac{d}{dx}(2x-1) = 2$. Since $f'(1^-) \ne f'(1^+)$, $f(x)$ is not differentiable at $x=1$.

Step 6: Conclusion

$f(x)$ is continuous but not differentiable at both $x=0$ and $x=1$.

Final Answer: f(x) is continuous but not differentiable, at x=0 and x=1.

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply the concepts of continuity and differentiability to a given function and determine its properties at specific points.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the concepts of continuity and differentiability, and how they relate to piecewise-defined functions involving absolute values.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question requires students to apply their knowledge of continuity and differentiability to a specific function, demonstrating a deeper understanding and application of the concepts rather than just recalling definitions or formulas.
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