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Question
Find the values of a and b such that the function defined by `f(x) = {(5, "," if x <= 2),(ax +b, "," if 2 < x < 10),(21, "," if x >= 10):}` is a continuous function.
Solution
`f(x) = {(5, "," if x <= 2),(ax +b, "," if 2 < x < 10),(21, "," if x >= 10):}`
Since f(x) = 5, f(x) = ax + b, f(x) is a continuous function at 21 times, f(x) is already a continuous function at x < 2, 2 < x < 10, x > 10.
If f(x) is continuous at x = 2, this implies:
`f(2) = lim_(x -> 2^+) f(x) = lim_(x -> 2^-) f(x)`
`=> 5 = a(2) + b = 5`
`=> 2a + b = 5` ...(1)
If f(x) is continuous at x = 10, this implies:
`f(10) = lim_(x -> 10^+) f(x) = lim_(x -> 10^-) f(x)`
`=> 21 = 21 = a(10) + b `
`=> 10a + b = 21` ...(2)
Subtracting equation (2) from (1),
`=> 8a = 16`
`=> a = 2`
`therefore (2) (2) + b = 5`
`=> b = 1`
That is, the function f(x) is continuous for the quantities a = 2, b = 1.
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