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Question
Find a and b if the following function is continuous at the point indicated against them.
`f(x) = x^2 + a` , for x ≥ 0
= `2sqrt(x^2 + 1) + b` , for x < 0 and
f(1) = 2 is continuous at x = 0
Solution
Since, f(x) = x2 + a, x ≥ 0
∴ f(1) = (1)2 + a
∴ 2 = 1 + a ...[∵ f(1) = 2]
∴ a = 1
Also f is continuous at x = 0
∴ `lim_(x→0^-) "f"(x) = lim_(x→0^+) "f"(x)`
∴ `lim_(x→0^-) (2sqrt(x^2 + 1) + "b") = lim_(x→0^+) (x^2 + "a")`
∴ `2sqrt(0^2 + 1) + "b" = 0^2 + 1`
∴ 2(1) + b = 1
∴ b = –1
∴ a = 1 and b = –1
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