Question

In the following, determine the value of constant involved in the definition so that the given function is continuou:   \[f\left( x \right) = \begin{cases}4 , & \text{ if } x \leq - 1 \\ a x^2 + b, & \text{ if }  - 1 < x < 0 \\ \cos x, &\text{ if }x \geq 0\end{cases}\]

Answer 1

Given: 

  \[f\left( x \right) = \begin{cases}4 , & \text{ if } x \leq - 1 \\ a x^2 + b, & \text{ if }  - 1 < x < 0 \\ \cos x, &\text{ if }x \geq 0\end{cases}\]

If  \[f\left( x \right)\] is continuous at x = −1 and 0, then 

\[\lim_{x \to - 1^-} f\left( x \right) = \lim_{x \to - 1^+} f\left( x \right) \text{ and }  \lim_{x \to 0^-} f\left( x \right) = \lim_{x \to 0^+} f\left( x \right)\]

\[\Rightarrow \lim_{h \to 0} f\left( - 1 - h \right) = \lim_{h \to 0} f\left( - 1 + h \right) \text{ and }  \lim_{h \to 0} f\left( - h \right) = \lim_{h \to 0} f\left( h \right) \]
\[ \Rightarrow \lim_{h \to 0} \left( 4 \right) = \lim_{h \to 0} \left( a \left( - 1 + h \right)^2 + b \right) \text{ and } \lim_{h \to 0} \left( a \left( - h \right)^2 + b \right) = \lim_{h \to 0} \left( \cos h \right)\]
\[ \Rightarrow 4 = a + b \text{ and }  b = 1\]
\[ \Rightarrow a = 3 \text{ and } b = 1\]