Question

For what values of a and b is the function

f(x) `{:(= "a"x + 2"b" + 18",",  "for"  x ≤ 0),(= x^2 + 3"a" - "b"",",  "for"  0 < x ≤ 2),(= 8x - 2",",  "for"  x > 2):}}` continuous for every x?

Answer 1

If f is continuous for every x, then f must be continuous at x = 0 and x = 2.

Continuity at x = 0

Since f is continuous at x = 0,

`lim_(x -> 0) "f"(x)` exists

∴ `lim_(x -> 0^+) "f"(x) = lim_(x -> 0^-) "f"(x)`

∴ `lim_(x -> 0) (x^2 + 3"a" - "b") =  lim_(x -> 0) ("a"x + 2"b" + 18)`

∴ 0 + 3a – b = 0 + 2b + 18

∴ 3a – 3b = 18

∴ a – b = 6   ...(1)

Continuity at x = 2

Since f is continuous at x = 2,

`lim_(x -> 2) "f"(x)` exists

∴ `lim_(x -> 2^+) "f"(x) = lim_(x -> 2^-) "f"(x)`

∴ `lim_(x -> 2) (8x - 2) =  lim_(x -> 2) (x^2 + 3"a" - "b")` 

∴ 8(2) – 2 = 4 + 3a – b

∴ 3a – b = 10

∴ 3a – (a – 6) = 10   ...[By (1)]

∴ 2a = 4

∴ a = 2

Substituting a = 2 in (1), we get,

∴ 2 – b = 6

∴ b = – 4

Hence, a = 2, b = – 4.