Question

If the function f(x) given by f(x)={3ax+b,ifx>011,ifx=15ax−2b,ifx<1 is continuous at x = 1, find the values of a and b.

Answer 1

The function f(x) given by,

{3ax + b, if x > 1

f(x) = { 11 , if x = 1

{ 5ax - 2b , if x < 1 , is continuous at x = 1.

To find : The values of a and b.

solution : concept : A function y = f(x) is continuous at x = a {where a belongs to domain of f(x)}, only if LHL of f(x) at x = a = RHL of f(x) at x = a = f(a)

here function f(x) is continuous at x = 1

so, LHL of f(x) at x = 1 = RHL of f(x) at x = 1 = f(1)

now LHL of f(x) at x = 1 = lim(x → 1¯) f(x)

= lim(x → 1¯) (5ax - 2b) = 5a × 1 - 2b = 5a - 2b

RHL of f(x) at x = 1 = lim(x → 1⁺) f(x)

= lim(x → 1⁺) (3ax + b) = 3a × 1 + b = 3a + b

and f(1) = 11 [ given ]

so, 5a - 2b = 3a + b = 11

5a - 2b = 11 .......(1)

3a + b = 11 .........(2)

multiplying 2 with eq (2) and adding eq (1) we get,

2(3a + b) + (5a - 2b) = 2 × 11 + 11

⇒6a + 2b + 5a - 2b = 33

⇒11a = 33

⇒a = 3

and b = 2

Therefore the values of a = 3 and b = 2