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Question
For what values of a and b is the function
f(x) `{:(= "a"x + 2"b" + 18",", "for" x ≤ 0),(= x^2 + 3"a" - "b"",", "for" 0 < x ≤ 2),(= 8x - 2",", "for" x > 2):}}` continuous for every x?
Solution
If f is continuous for every x, then f must be continuous at x = 0 and x = 2.
Continuity at x = 0
Since f is continuous at x = 0,
`lim_(x -> 0) "f"(x)` exists
∴ `lim_(x -> 0^+) "f"(x) = lim_(x -> 0^-) "f"(x)`
∴ `lim_(x -> 0) (x^2 + 3"a" - "b") = lim_(x -> 0) ("a"x + 2"b" + 18)`
∴ 0 + 3a – b = 0 + 2b + 18
∴ 3a – 3b = 18
∴ a – b = 6 ...(1)
Continuity at x = 2
Since f is continuous at x = 2,
`lim_(x -> 2) "f"(x)` exists
∴ `lim_(x -> 2^+) "f"(x) = lim_(x -> 2^-) "f"(x)`
∴ `lim_(x -> 2) (8x - 2) = lim_(x -> 2) (x^2 + 3"a" - "b")`
∴ 8(2) – 2 = 4 + 3a – b
∴ 3a – b = 10
∴ 3a – (a – 6) = 10 ...[By (1)]
∴ 2a = 4
∴ a = 2
Substituting a = 2 in (1), we get,
∴ 2 – b = 6
∴ b = – 4
Hence, a = 2, b = – 4.
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