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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability1 Mark
Question
Find the values of a and b such that the function defined by
f(x) = $ \left\{ \begin{array} { c c } { 5 , } & {if\ x \leq 2 } \\ { a x + b , } & {if\ 2 < x < 10 } \\ { 21 , } & {if\ x \geq 10 } \end{array} \right.$ is a continuous function.
✓

Answer

According to the question, f(x) = $ \left\{ \begin{array} { c c } { 5 , } & { x \leq 2 } \\ { a x + b , } & { 2 < x < 10 } \\ { 21 , } & { x \geq 10 } \end{array} \right.$ is a continuous function. So, it is continuous at x = 2 and at x = 10.
$\therefore LHL_{x = 2} = RHL_{x = 2} = f(2)$ .....(i)
and $LHL_{x = 10} = RHL_{x = 10} = f(10)$ .....(ii)
Now, let us calculate LHL and RHL at x = 2.
LHL = $ \lim _ { x \rightarrow 2 ^ { - } } f ( x ) = \lim _ { x \rightarrow 2 ^ { - } } 5 = 5$
and RHL = $ \lim _ { x \rightarrow 2 ^ { + } } f ( x ) = \lim _ { x \rightarrow 2 ^ { + } } ( a x + b )$
$ = \lim _ { h \rightarrow 0 } \{ a ( 2 + h ) + b \} = \lim _ { h \rightarrow 0 } ( 2 a + a h + b )$
= 2a + b
$ \Rightarrow$ 2a + b = 5.........(iii)
Now, we have to find LHL and RHL at x = 10.
LHL = $ \lim _ { x \rightarrow 10 ^ { - } } f ( x ) = \lim _ { x \rightarrow 10 ^ { - } } ( a x + b )$
$ = \lim _ { h \rightarrow 0 } [ a ( 10 - h ) + b ]$
$ = \lim _ { h \rightarrow 0 } ( 10 a - a h + b )$
$ \Rightarrow$LHL = 10a + b
and RHL = $ \lim _ { x \rightarrow 10 ^ { + } } f ( x ) = \lim _ { x \rightarrow 10 ^ { + } } 21$ = 21
Now, from Eq. (ii), we have
LHL= RHL
$ \Rightarrow$ $10 a + b = 21$ ......(iv)
Subtracting Equation (iv) from Equation (iii),
$\Rightarrow - 8a = -16$
$\Rightarrow$ $a = 2$
Putting a = 2 in Equation (iv),
10 (2) + b = 21
$ \Rightarrow$ $b = 1$
$\therefore$ $a = 2 \ and\ b = 1$

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