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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY2 Marks
Question
Find the relationship between a and b so that the function f defined by
$\text{f(x)}= \begin{cases}\text{ax} + 1, \text{if}\ \text{x} \leq3\\ \text{bx} + 3, \text{if}\ \text{x} > 3\end{cases}$
is continuous at x = 3.
✓

Answer

$\text{f(x)}= \begin{cases}\text{ax} + 1, \text{if}\ \text{x} \leq3\\ \text{bx} + 3, \text{if}\ \text{x} > 3\end{cases}$$^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{3}^{-}}\text{f(x)} = ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{3}^{-}}(\text{ax + 1}) = \text{3a + 1}$
$^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{3}^{+}}\text{f(x)} = ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{3}^{+}}(\text{bx + 3}) = \text{3b + 3}$
Also f(3) = 3a + 1 Since f is continuous at x = 3 $\therefore \ ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{3}^{-}}\text{f(x)} = ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{3}^{+}}\text{f(x }) = \text{f(3)}$ $\therefore$ 3a + 1 = 3b + 3 = 3a + 1 $\therefore$ 3a = 3b + 2 $\Rightarrow\ \text{a} = \text{b}+\frac{2}{3}$, which is required relation.

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