Let \(f: R \rightarrow R\) be the function defined by \(f(x)=\left\{\begin{array}{cl}5, & \text { if } x…
Let \(f: R \rightarrow R\) be the function defined by \(f(x)=\left\{\begin{array}{cl}5, & \text { if } x \leq 1 \\ a+b x, & \text { if } 1 < x < 3 \\ b+5 x, & \text { if } 3 \leq x < 5 \\ 30, & \text { if } x \geq 5\end{array}\right.\) then \(f\) is
continuous if \(a=5\) and \(b=5\)
continuous if \(a=0\) and \(b=5\)
continuous if \(a=-5\) and \(b=10\)
not continuous for any values of \(a\) and \(b\)
Solution
Given function \(f: R \rightarrow R\), such that
\(f(x)=\left[\begin{array}{cl}
5, & \text { if } x \leq 1 \\
a+b x, & \text { if } 1 < x < 3 \\
b+5 x, & \text { if } 3 \leq x < 5 \\
30, & \text { if } x \geq 5
\end{array}\right.\)
If \(f\) is continuous at \(x=1\), then
\(a+b=5\) ...(i)
If \(f\) is continuous at \(x=3\), then
\(\Rightarrow \quad \begin{gathered}
a+3 b=b+15 \\
a+2 b=15 \quad \ldots (ii)
\end{gathered}\)
and if \(f\) is continuous at \(x=5\), then
\(b+25=30 \Rightarrow b=5\) ...(iii)
From Eqs. (ii) and (iii), we get
\(a=5\) ...(iv)
but \(a=5\) and \(b=5\) doesn't satisfy the Eq. (i). so, \(f: R \rightarrow R\) is not continuous for any values of \(a\) and \(b\).
Hence, option (4) is correct.