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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability4 Marks
Question
If given function is continuous at $x=1$, then find $a$ and $b$.$
f(x)=\left\{\begin{array}{cl}
3 a x+b, & \text { if } x>1 \\
11, & \text { if } x=1 \\
5 a x-2 b, & \text { if } x<1
\end{array}\right.
$
✓

Answer

Given function
$
f(x)=\left\{\begin{array}{cl}
3 a x+b, & \text { if } x>1 \\
11, & \text { if } x=1 \\
5 a x-2 b, & \text { if } x<1
\end{array}\right.
$
value of L.H.L.
$
\begin{aligned}
\lim _{x \rightarrow 1^{-}} f(x) & =\lim _{x \rightarrow 1^{-}}[5 a x-2 b] \\
& =\lim _{h \rightarrow 0}[5 a(1-h)-2 b] \\
& =\lim _{h \rightarrow 0}[5 a-2 b-5 a h] \\
& =5 a-2 b \\
\therefore \quad \lim _{x \rightarrow 1^{-}} f(x) & =5 a-2 b .
\end{aligned}
$
value of R.H.L.$
\begin{array}{l}
\qquad \begin{aligned}
& =\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1^{+}}(3 a x+b) \\
& =\lim _{h \rightarrow 0}[3 a(1+h)+b] \\
& =\lim _{h \rightarrow 0}(3 a+b+3 a h)=3 a+b \\
\therefore \quad \lim _{x \rightarrow l^{+}} & =3 a+b \\
\text { and } \quad f(1) & =11
\end{aligned}
\end{array}
$
because function is continuous at $x=1$$
\begin{array}{rlrl} 
f(1) & =\lim _{h \rightarrow 0} f(1+h)=\lim _{h \rightarrow 0} f(1-h) \\
\Rightarrow & 11=3 a+b=5 a-2 b
\end{array}
$

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