If the function \( f(x)=\begin{cases}x^{2}, & x\le c\\ ax+b, & x\gt c\end{cases} \) is differentiable at \( x=c \), then the relation between \( b \) and \( c \) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( b=-c^{2} \)
A function differentiable at x = c is continuous there, and its left and right derivatives are equal.
Continuity at x = c:
\( \displaystyle c^{2}=ac+b \)
Equal derivatives at x = c:
\( \displaystyle 2c=a \)
Substituting a = 2c in the first equation,
\( \displaystyle c^{2}=2c^{2}+b \)
\( \displaystyle b=-c^{2} \)