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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.
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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
⇒ 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 = a(10) + b
⇒ 10a + b = 21 ...(2)
Subtracting equation (2) from (1),
⇒ 8a = 16
⇒ a = `16/8`
⇒ a = 2
Put a = 2 in equation (1)
⇒ 2(2) + b = 5
⇒ 4 + b = 5
⇒ b = 5 − 4
⇒ b = 1
That is, the function f(x) is continuous for the quantities a = 2, b = 1.
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