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Question
If the function f is continuous at x = 0
Where f(x) = 2`sqrt(x^3 + 1)` + a, for x < 0,
= `x^3 + a + b, for x > 0
and f (1) = 2, then find a and b.
Solution
Consider.
`lim_(x->0^-) f(x) = lim_(x->0^-) [2sqrt(x^3 + 1) + a]`
= 2 + a.........(1)
`lim_(x->0^+) f(x) = lim_(x->0^+) [x^3 + a +b]`
= a + b......(ii)
f(o) = a + b........(iii)
Since f is continuous at x = 0
`lim_(x->0^-) f(x) = lim_(x->0^+) f(x) = f(0)`
2 + a = a + b
b = 2
Also f(1) = 2
f(1) = 13 + a + b
2 = 1 + a + b
1 = a + 2
a = -1
a = -1 and b = 2
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