Question

Find the values of `a` and ` b` such that the function by:

`f(x) = {{:(5",", if  x ≤ 2),(ax + b",", if 2 < x < 10),(21",", if x ≥ 10):}`

is a continuous function.

विकल्प

• 0, 1

• 1, 1

• 2, 1

• 1, 2

Answer 1

2, 1

Explanation:

At `x` = 2, L.H.L = `lim_(x -> 2^-) (5)` = 5

`f(2)` = 5

R.H.L = `lim_(x -> 2^+) (ax + b) = 2a + b`

`f` is continuous at `x = 2, 2a + b` = 5  ......(1)

At `x` = 10, L.H.L = `lim_(x -> 10^-) f(x) = lim_(x -> 10^-) (ax + b) = 10a + b`

R.H.L = `lim_(x -> 10^+) f(x) = lim_(x -> 10^+)`

`f` is continuous at `x` = 10 if 10`a + b` = 21  ......(2)

Subtracting (1) and (2),

`8a = 21 - 5 - 16`,     ∴ `a` = 2

From (1), `2 xx 2 + b` = 5   ∴  `b` = 1

Hence `a` = 2, `b` = 1.