Differentiability
Define f(x) = x^2 for x \le 1 and f(x) = ax + b for x > 1; find a, b making f differentiable everywhere, then a+b.
Select the correct option:
Solution
1
Differentiability of a piecewise function at the join x = 1 demands two matching conditions: the values must agree for continuity, and the one-sided derivatives must agree for a common tangent. Continuity at x = 1 gives 1^2 = a(1) + b, so a + b = 1. Differentiability requires the left derivative 2x|_{x=1} = 2 to equal the right derivative a, so a = 2. Substituting back, 2 + b = 1, hence b = -1, and the pair (a,b) = (2,-1) makes f smooth across the junction. Therefore a + b = 1, exactly the continuity equation. Option 0 satisfies neither matching condition consistently. Option 2 uses only the derivative match a = 2 and mislabels it as the sum. Option 3 adds the two derivatives instead of using the continuity sum. The governing JEE pattern is the dual value-and-slope matching for differentiable piecewise functions. Plausibility check: with a = 2, b = -1 the right piece is 2x - 1, whose value at x = 1 is 1 and whose slope is 2, matching the parabola's value 1 and slope 2 precisely, so the graph joins smoothly and a + b = 1 holds.
🔒 Solution Hidden from View
Submit your answer to unlock the detailed step-by-step solution.
More differentiability Practice Questions
Consider h(x) = |x - 1| + |x + 1| on the real line; at how many points does h fail to be differentia...
Consider h(x) = |x - 1| + |x + 1| on the real line; at how many points does h fail to be differentia...
Let p(x) = x^2 \sin(1/x) for x \neq 0 and p(0) = 0; which statement about p at the origin is correct...
Let p(x) = x^2 \sin(1/x) for x \neq 0 and p(0) = 0; which statement about p at the origin is correct...
If y = x^x for x > 0, then the derivative dy/dx evaluated through logarithmic differentiation equals...
If y = x^x for x > 0, then the derivative dy/dx evaluated through logarithmic differentiation equals...
Assertion: every differentiable function is continuous. Reason: differentiability requires the diffe...
Assertion: every differentiable function is continuous. Reason: differentiability requires the diffe...
Let q(x) = \max{x, x^3} for real x; determine the set of points where q is not differentiable.
Let q(x) = \max{x, x^3} for real x; determine the set of points where q is not differentiable.
About This Question
- Subject
- mathematics
- Chapter
- limit, continuity and differentiability
- Topic
- differentiability
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
1
Differentiability of a piecewise function at the join x = 1 demands two matching conditions: the values must agree for continuity, and the one-sided derivatives must agree for a common tangent. Continuity at x = 1 gives 1^2 = a(1) + b, so a + b = 1. Differentiability requires the left derivative 2x|_{x=1} = 2 to equal the right derivative a, so a = 2. Substituting back, 2 + b = 1, hence b = -1, and the pair (a,b) = (2,-1) makes f smooth across the junction. Therefore a + b = 1, exactly the continuity equation. Option 0 satisfies neither matching condition consistently. Option 2 uses only the derivative match a = 2 and mislabels it as the sum. Option 3 adds the two derivatives instead of using the continuity sum. The governing JEE pattern is the dual value-and-slope matching for differentiable piecewise functions. Plausibility check: with a = 2, b = -1 the right piece is 2x - 1, whose value at x = 1 is 1 and whose slope is 2, matching the parabola's value 1 and slope 2 precisely, so the graph joins smoothly and a + b = 1 holds.
This medium difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of differentiability. It appeared in the 2025 exam.
Looking for more practice? Explore all mathematics questions or browse limit, continuity and differentiability questions on RankGuru.