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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
In the following, find the value of the constant k so that the given function is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\text{k}+1,&\text{if}\text{ x}\leq5\\3\text{x}-5,&\text{if}\text{ x}>5\end{cases}\text{at x} =5$
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Answer

 $\text{f(x)}=\begin{cases}\text{k}+1,&\text{if}\text{ x}\leq5\\3\text{x}-5,&\text{if}\text{ x}>5\end{cases}\text{at x} =5$

We have given that function is continuous at x = 5

$\therefore\ \text{LHL}=\text{RHL}=\text{f}(5)\ ....(\text{i})$

$\text{f}(5)=5\text{k}+1$

$\text{LHL}=\lim_\limits{\text{x}\rightarrow5^+}\text{f(x)}=\lim_\limits{\text{h}\rightarrow0}(5+\text{h})$

$=\lim_\limits{\text{h}\rightarrow0}3(5+\text{h})-5=10$

Thus, using (i) we get,

$5\text{k}+1=10$

$5\text{k}=9$

$\text{k}=\frac{9}{5}$ 

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