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.

Answer 1

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