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Differentiability — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferentiability5 Marks
Question
Find the values of a and b so that the function $\text{f(x)}\begin{cases}\text{x}^2+3\text{x}+\text{a}, & \text{if x}\leq1\\\text{bx}+2, & \text{if x} > 1\end{cases}$ is differentiable at each $\text{x}\in\text{R}.$
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Answer

It is given that the function is differentiable at each $\text{x}\in\text{R}$ and every differentiable function is continuous.
Therefore,
Given: $\text{f(x)}=\begin{Bmatrix}\text{x}^2+3\text{x}+\text{a}, & \text{if x}\leq1\\\text{bx}+2, & \text{if x} > 1 \end{Bmatrix}$
Therefore,
f(x) is continuous at x = 1.
Therefore,
$\lim_\limits{\text{x}\rightarrow1^{-}}\text{f(x)}=\lim_\limits{\text{x}\rightarrow1^{+}}\text{f(x)}=\text{f}(1)$
Implies that $\lim_\limits{\text{x}\rightarrow1}\text{x}^2+3\text{x}+\text{a}=\lim_\limits{\text{x}\rightarrow1}\text{bx}+2=\text{a}+4$ [Using def. of f(x)]
Implies that a + 4 = b + 2 = a + 4 ......(1)
Since, f(x) is differentiable at x = 1. Therefore,
(LHL at x = 1) = (RHL at x = 1)
$\lim_\limits{\text{x}\rightarrow1^{-}}\frac{\text{f(x)}-\text{f}(1)}{\text{x}-1}=\lim_\limits{\text{x}\rightarrow1^{+}}\frac{\text{f(c)}-\text{f}(1)}{\text{x}-1}$
Implies that $\lim_\limits{\text{x}\rightarrow1}\frac{\text{x}^2+3\text{x}+\text{a}-\text{a}-4}{\text{x}-1}=\lim_\limits{\text{x}\rightarrow1}\frac{\text{bx}+2-4-\text{a}}{\text{x}-1}$ [Using def. of f(x)]
Implies that $\lim_\limits{\text{x}\rightarrow1}\frac{(\text{x}+4)(\text{x-1})}{\text{x}-1}=\lim_\limits{\text{x}\rightarrow1}\frac{\text{bx}-2-\text{a}}{\text{x}-1}$
Implies that $\lim_\limits{\text{x}\rightarrow1}\frac{(\text{x}+4)(\text{x}-1)}{\text{x}-1}=\lim_\limits{\text{x}\rightarrow1}\frac{\text{bx}-\text{b}}{\text{x}-1}$ [Using (1)]
Implies that $\lim_\limits{\text{x}\rightarrow1}\frac{(\text{x}+4)(\text{x}-1)}{\text{x}-1}=\lim_\limits{\text{x}\rightarrow1}\frac{\text{b}(\text{x}-1)}{\text{x}-1}$
Implies that 5 = b
From (1), we have
a + 4 = b + 2
Implies that a + 4 = 5 + 2
Implies that a = 7 - 4
Implies that a = 3
Hence, a = 3, b = 5.

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