Question
(JEE Main 2024 PYQ — paraphrased) Find the values of and such that the function
is differentiable at .
Solution — Step by Step
For differentiability at , the function must first be continuous there. Set:
Left limit: . Right limit: . Equate:
Left derivative at : for , so .
Right derivative at : for , so .
For differentiability:
From (ii), . Substitute into (i): , which is impossible.
So there are no values of that make differentiable at .
If the question instead asks for continuity, any with works (e.g., ).
Why This Works
Differentiability at a point requires:
- Continuity at that point.
- Equal left- and right-hand derivatives.
For piecewise functions, both conditions give equations relating the parameters. If the system is consistent, you get unique parameter values; if inconsistent, the function cannot be made differentiable.
In this problem, the right branch has a “+3” extra constant that creates a jump that cannot be reconciled with equal slopes — geometrically, the right line is shifted up, so even with equal slopes, the curves don’t meet.
For JEE Main piecewise differentiability questions, always set continuity first, then derivatives. Both conditions are needed — neither is sufficient alone.
Alternative Method
Geometric: the left branch is a line through with slope . The right is a line through with slope . For the lines to merge into one differentiable curve at , both their values and slopes must agree at — same conditions.
Common Mistake
Students apply only the differentiability condition and don’t check continuity. They report ”, any value works” — which is wrong, because continuity additionally requires . The full check reveals the inconsistency.