Differentiability
Let q(x) = \max{x, x^3} for real x; determine the set of points where q is not differentiable.
Select the correct option:
Solution
{−1,0,1}
The maximum of two smooth functions can fail to be differentiable only where the two graphs cross, because at such crossings the active branch switches and slopes may disagree. Solve x = x^3, i.e. x(x^2 - 1) = 0, giving crossing points x = -1, 0, 1. Between these, q equals whichever of x or x^3 is larger, each smooth on its own. At every crossing the slopes of x and x^3 differ: at x = 0 the slopes are 1 and 0, at x = \pm 1 the slopes are 1 and 3, so all three are genuine corners. Hence the non-differentiable set is {-1, 0, 1}. Option {0} captures only one crossing and ignores \pm 1. Option {-1, 1} omits the origin where slopes 1 and 0 clash. Option {-1, 0, 1, 2} invents an extra point where no crossing occurs. The governing JEE pattern is differentiability of a max of functions at their intersection points. Plausibility check: at each listed point the left and right derivatives are unequal, the defining signature of a corner, and there are exactly three real solutions of x = x^3, matching the set size of three.
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About This Question
- Subject
- mathematics
- Chapter
- limit, continuity and differentiability
- Topic
- differentiability
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
{−1,0,1}
The maximum of two smooth functions can fail to be differentiable only where the two graphs cross, because at such crossings the active branch switches and slopes may disagree. Solve x = x^3, i.e. x(x^2 - 1) = 0, giving crossing points x = -1, 0, 1. Between these, q equals whichever of x or x^3 is larger, each smooth on its own. At every crossing the slopes of x and x^3 differ: at x = 0 the slopes are 1 and 0, at x = \pm 1 the slopes are 1 and 3, so all three are genuine corners. Hence the non-differentiable set is {-1, 0, 1}. Option {0} captures only one crossing and ignores \pm 1. Option {-1, 1} omits the origin where slopes 1 and 0 clash. Option {-1, 0, 1, 2} invents an extra point where no crossing occurs. The governing JEE pattern is differentiability of a max of functions at their intersection points. Plausibility check: at each listed point the left and right derivatives are unequal, the defining signature of a corner, and there are exactly three real solutions of x = x^3, matching the set size of three.
This hard difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of differentiability. It appeared in the 2025 exam.
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