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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

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#### Solution

Since, f(x) = x^{2} + 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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