Question

For what values of a and b is the function

f(x) = ax + 2b + 18    for x ≤ 0

= x² + 3a − b            for 0 < x ≤ 2

= 8x – 2                     for x > 2,

continuous for every x ?

Answer 1

Function f is continuous for every x.
∴ Function f is continuous at x = 0 and x = 2
As f is continuous at x = 0.
∴ `lim_(x→0^-) "f"(x) = lim_(x→0^+) "f"(x)`

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

∴ a(0) + 2b + 18 = (0)² + 3a - b
∴ 3a - 3b = 18
∴ a – b = 6       ...(i)
Also, Function f is continous at x = 2
∴ `lim_(x→2^-) "f"(x) = lim_(x→2^+) "f"(x)`

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

∴ (2)² + 3a – b = 8 (2) – 2
∴ 4 + 3a – b = 14
∴ 3a – b = 10        …(ii)
Subtracting (i) from (ii), we get
2a = 4
∴ a = 2
Substituting a = 2 in (i), we get
2 - b = 6
∴ b = – 4
∴ a = 2 and b = – 4