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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Find the values of a and b such that the function defined by
$\text{f(x)}=\begin{cases}5,&\text{if}\ \text{x}\leq{2}\\\text{ax} + \text{b},& \text{if}\ 2<\text{x}<10\\21,&\text{if}\ \text{x}\geq10\end{cases}$
is a continuous function.
✓

Answer

$\text{f(x)}=\begin{cases}5,&\text{if}\ \text{x}\leq{2}\\\text{ax} + \text{b},& \text{if}\ 2<\text{x}<10\\21,&\text{if}\ \text{x}\geq10\end{cases}$
$\therefore$ f is continuous function
$\therefore$ f is continuous at x = 2 and x = 10
$\therefore$ f is right continuous at x = 2 and left continuous at x = 10.
$\therefore\ ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{2}^{+}}\text{f(x)} = \text{f}(2)\Rightarrow 2\text{a} + \text{b} = 5\ ...{(\text i)}$
$^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{10}^{-}}\text{f(x)} = \text{f}(10)\Rightarrow 10\text{a} + \text{b} = 21\ ...{(\text {ii})}$
Subtracting (1) from(2), we get,
8a = 16 or a = 2
$\therefore$ from (1), 4 + 6 = 5 ⇒ b = 1
$\therefore$ we have a = 2, b = 1

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