Differentiability in Assertion-Reasoning Questions: A Strategic Approach
Master the art of solving Assertion-Reasoning questions on Differentiability with proven strategies and common patterns from JEE Main.
Why Assertion-Reasoning Questions Matter
Assertion-Reasoning questions in JEE Main test conceptual depth and logical reasoning simultaneously. In Differentiability, these questions account for 15-20% of calculus marks and follow predictable patterns.
- High scoring potential - 4 marks for correct logical analysis
- Conceptual testing - Tests understanding beyond computation
- Time-efficient - Once mastered, faster to solve than numerical problems
- Common patterns - 80% questions follow 4-5 standard templates
4-Step Strategy for Assertion-Reasoning Questions
Step 1: Analyze Assertion Independently
Check if the assertion statement is True or False without considering the reason.
Step 2: Analyze Reason Independently
Verify if the reason statement is mathematically correct on its own.
Step 3: Check Connection
Determine if the reason correctly explains the assertion when both are true.
Step 4: Apply Standard Cases
Use common differentiability patterns: absolute functions, piecewise functions, composite functions.
Problem 1: Absolute Value Differentiability
Assertion (A): The function $f(x) = |x-1| + |x-2|$ is not differentiable at $x=1$ and $x=2$.
Reason (R): The sum of two non-differentiable functions at a point is non-differentiable at that point.
Strategic Analysis:
Step 1 - Check Assertion: $f(x) = |x-1| + |x-2|$
• At $x=1$: LHD = $-1$, RHD = $1$ ⇒ Not differentiable ✅
• At $x=2$: LHD = $-1$, RHD = $1$ ⇒ Not differentiable ✅
Step 2 - Check Reason: "Sum of two non-differentiable functions is non-differentiable"
• Counterexample: $f(x) = |x|$ and $g(x) = -|x|$ are both non-differentiable at 0
• But $f(x) + g(x) = 0$ is differentiable everywhere ❌
Step 3 - Conclusion: Assertion is TRUE, Reason is FALSE
Answer: (C) A is true but R is false
Problem 2: Product Function Differentiability
Assertion (A): If $f(x) = x^2|x|$, then $f(x)$ is differentiable at $x=0$.
Reason (R): The product of a differentiable function and a non-differentiable function is always non-differentiable.
Strategic Analysis:
Step 1 - Check Assertion: $f(x) = x^2|x| = x^2 \cdot |x|$
• For $x \geq 0$: $f(x) = x^3$ ⇒ $f'(x) = 3x^2$
• For $x < 0$: $f(x) = -x^3$ ⇒ $f'(x) = -3x^2$
• At $x=0$: LHD = $0$, RHD = $0$ ⇒ Differentiable ✅
Step 2 - Check Reason: "Product of differentiable and non-differentiable is always non-differentiable"
• Counterexample: $g(x) = x$ (differentiable) and $h(x) = |x|$ (non-differentiable at 0)
• But $g(x) \cdot h(x) = x|x|$ is differentiable at 0 ❌
Step 3 - Conclusion: Assertion is TRUE, Reason is FALSE
Answer: (C) A is true but R is false
Problem 3: Continuity vs Differentiability
Assertion (A): If a function is differentiable at a point, it must be continuous at that point.
Reason (R): Differentiability implies continuity, but continuity does not imply differentiability.
Strategic Analysis:
Step 1 - Check Assertion: "Differentiable ⇒ Continuous"
• Fundamental theorem: If $f'(c)$ exists, then $\lim_{x\to c} f(x) = f(c)$ ✅
Step 2 - Check Reason: "Differentiable ⇒ Continuous, but Continuous ⇏ Differentiable"
• Example: $f(x) = |x|$ is continuous at 0 but not differentiable ✅
Step 3 - Check Connection: Does R correctly explain A?
• Yes, R provides the complete relationship between differentiability and continuity ✅
Step 4 - Conclusion: Both TRUE and R is correct explanation of A
Answer: (A) Both A and R are true and R is correct explanation
🚀 Quick Solving Strategies
Common True Patterns:
- Differentiable ⇒ Continuous (always)
- Derivative exists ⇒ LHD = RHD
- Polynomials are differentiable everywhere
- Composite of differentiable functions is differentiable
Common False Patterns:
- Continuous ⇒ Differentiable (false)
- Sum of non-differentiable functions is non-differentiable (false)
- Product with non-differentiable function is non-differentiable (false)
- Absolute value functions are non-differentiable at vertex (true)
Problems 4-8 Available in Full Version
Includes 5 more essential JEE Main Assertion-Reasoning problems with strategic analysis
📝 Quick Self-Test
Try these assertion-reasoning patterns to test your understanding:
1. Assertion: $f(x) = x|x|$ is differentiable at $x=0$
Reason: Product of differentiable and non-differentiable functions is always differentiable
2. Assertion: $f(x) = |x-1|(x^2+1)$ is not differentiable at $x=1$
Reason: Absolute value functions are non-differentiable at their vertex points
3. Assertion: If $f'(c)$ exists, then $f(x)$ is continuous at $x=c$
Reason: Existence of derivative implies the function is smooth at that point
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