Question:
The function f is defined by $f(x)=\begin{cases}x^{2}+3x+a,& if~x\le1\\ bx+2,& if~x>1\end{cases}$ is derivable for every x. Then the value of a is}
The function f is defined by $f(x)=\begin{cases}x^{2}+3x+a,& if~x\le1\\ bx+2,& if~x>1\end{cases}$ is derivable for every x. Then the value of a is}
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For piecewise functions to be "derivable," solve for slope ($b$) first, then solve for continuity ($a$).
Updated On: Jul 3, 2026
- 3
- 5
- -3
- -6
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The Correct Option is A
Solution and Explanation
Step 1: Concept If a function is derivable (differentiable), it must be continuous and its derivatives must match at the boundary ($x=1$).
Step 2: Meaning Differentiability check: $f'(x) = 2x+3$ for $x < 1$ and $f'(x) = b$ for $x > 1$. Matching derivatives at $x=1$: $2(1)+3 = b \Rightarrow b = 5$.
Step 3: Analysis Continuity check: $f(1)$ from left $= (1)^2+3(1)+a = 4+a$. $f(1)$ from right $= b(1)+2 = 5+2 = 7$.
Step 4: Conclusion Set left and right limits equal: $4+a = 7 \Rightarrow a = 3$. Final Answer: (A)
Step 2: Meaning Differentiability check: $f'(x) = 2x+3$ for $x < 1$ and $f'(x) = b$ for $x > 1$. Matching derivatives at $x=1$: $2(1)+3 = b \Rightarrow b = 5$.
Step 3: Analysis Continuity check: $f(1)$ from left $= (1)^2+3(1)+a = 4+a$. $f(1)$ from right $= b(1)+2 = 5+2 = 7$.
Step 4: Conclusion Set left and right limits equal: $4+a = 7 \Rightarrow a = 3$. Final Answer: (A)
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