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

Answer 1

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