Question

In the following, determine the value of constant involved in the definition so that the given function is continuou:  \[f\left( x \right) = \begin{cases}5 , & \text{ if }  & x \leq 2 \\ ax + b, & \text{ if } & 2 < x < 10 \\ 21 , & \text{ if }  & x \geq 10\end{cases}\]

Answer 1

Given: 

  \[f\left( x \right) = \begin{cases}5 , & \text{ if }  & x \leq 2 \\ ax + b, & \text{ if } & 2 < x < 10 \\ 21 , & \text{ if }  & x \geq 10\end{cases}\]

If  \[f\left( x \right)\]  is continuous at x = 2 and 10,  then 

\[\lim_{x \to 2^-} f\left( x \right) = \lim_{x \to 2^+} f\left( x \right) \text{ and }  \lim_{x \to {10}^-} f\left( x \right) = \lim_{x \to {10}^+} f\left( x \right)\]

\[\Rightarrow \lim_{h \to 0} f\left( 2 - h \right) = \lim_{h \to 0} f\left( 2 + h \right) \text{ and } \lim_{h \to 0} f\left( 10 - h \right) = \lim_{h \to 0} f\left( 10 + h \right)\]
\[ \Rightarrow \lim_{h \to 0} \left( 5 \right) = \lim_{h \to 0} \left[ a\left( 2 + h \right) + b \right] \text{ and } \lim_{h \to 0} \left[ a\left( 10 - h \right) + b \right] = \lim_{h \to 0} \left( 21 \right)\]
\[ \Rightarrow 5 = 2a + b . . . \left( 1 \right) \text{ and }  10a + b = 21 . . . \left( 2 \right)\]
\[\text{ On solving eqs }  . \left( 1 \right) \text{ and }  \left( 2 \right), \text{ we get } \]
\[a = 2 \text{ and } b = 1\]