Limits, Continuity & Differentiability
Continuity And Differentiability
nta_abhyas_2025
Grade 12
Question:
If $f(x) = \begin{cases} 2x^2 + 3 & x > 3 \\ ax^2 + bx + 1 & x \leq 3 \end{cases}$ is differentiable everywhere, then $a$ is equal to
5
\frac{5}{9}
1
\frac{11}{9}
Step-by-Step Solution
Key Concept: Continuity requires left and right limits equal the function value; differentiability requires equal derivatives
For continuity at $x = 3$: $\lim_{x \to 3^+} f(x) = \lim_{x \to 3^-} f(x) = f(3)$. From the right: $\lim_{x \to 3^+} 4x = 12$. From the left: $f(3) = 2a(3) + b = 6a + b = 12$ ... (1). For differentiability at $x = 3$: $f'(3^+) = 4$ and $f'(3^-) = 2a$, so $2a = 4$, giving $a = 2$. Substituting in (1): $12 + b = 12$, so $b = 0$. However, solving the system $9a + 3b = 20$ and $6a + b = 12$ yields $a = \frac{8}{9}, b = \frac{4}{3}$.
Correct Answer: 1