Question

Find the constant b that makes g continuous on `(- oo, oo)`.

`g(x) = {{:(x^2 - "b"^2,"if"  x < 4),("b"x + 20,  "if"  x ≥ 4):}`

Answer 1

`g(x) = {{:(x^2 - "b"^2,"if"  x < 4),("b"x + 20,  "if"  x ≥ 4):}`

Given g is continuous on R.

∴ g(x) is continuous at x = 4.

`lim_(x -> 4^-) g(x) =  lim_(x -> 4^+) g(x)`

`lim_(x ->4^-) (x^2 - "b"^2) =  lim_(x -> 4^+) ("b"x + 20)`

4² – b² = b × 4 + 20

16 – b² = 4b + 20

b² + 4b + 20 – 16 = 0

b² + 4b + 4 = 0

(b + 2)² = 0

b + 2 = 0 ⇒ b = – 2