Continuity and Differentiability
Differentiability of piecewise function
nta_pyq_2025_apr
Grade 12
Question:
{ ⎪ ⎪ ⎪$(1 + ax)$$1/x$,$x < 0$$1 + b$,$x = 0$Let f$(x) = 1/2$$(x+4) -2$⎪ ⎪ ⎪ ,$1/3$$(x+c) -2$be continuous at$x = 0.$Then e bc is equal to a
Step-by-Step Solution
Key Concept: Break the function at its$formula-changing$points and compare$one-sided$limits or derivatives there.
f (0$) = e - lim$x$\to$0$x = e$a (3) f$(0) = 1 + b$1 1 2$\sqrtx$+4$2(2) + f$(0 ) = = 2 2$1 - 1 - (x + c)$3$\cdot$c 3 3 3 3$2/3 = c$4 Also at$x = 0$;$1/3$$c = 2$$\Rightarrow$$c = 8$So f (0 ) = + 3 4 (8)$2/3 = 3$Now,$e = b + 1 = 3$a a e . b$\cdot$$c = 3$$\cdot$2$\cdot$$8 = 48$
Correct Answer: 3