Question

# Find the values of $$a$$ and $$b$$ such that the function defined by $$f(x) = \left\{\begin{matrix} 5,& if & x\leq 2\\ ax + b, & if & 2 < x < 10\\ 21, & if & x \geq 10\end{matrix}\right.$$ is a continuous function.

Solution

## $$f(x) = \left\{\begin{matrix} 5,& if & x\leq 2\\ ax + b, & if & 2 < x < 10\\ 21, & if & x \geq 10\end{matrix}\right.$$If the function $$f(x)$$ is a continuous function, then$$f(2) = f(2^+)$$ and $$f(10^{-}) = f(10)$$$$I: f(2) = f(2^+)$$$$5 = 2a+b$$ ... (i)$$II: f(10^{-}) = f(10)$$$$10a+b = 21$$ ...(ii)from (i) and (ii), we get$$a = 2, b = 1$$Mathematics

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