Question

For what value of `k` the following function is continuous at the indicated point

`f(x) = {{:(kx^2",", if x ≤ 2),(3",", if x > 2):}` at x = 2

Options

• 4

• `2/4`

• `3/4`

• `4/4`

Answer 1

`3/4`

Explanation:

L.H.L = `lim_(x -> 2^-) (kx^2)`, putting `x = 2 - h`

= `lim_(h -> 0) k (2 - h)^2 - 4k`

`f(2) = k, 2^2 = 4k`

R.H.L = `lim_(x -> 2^+) f(x)` = 3

`f` is continuous if L.H.L = R.H.L = `f(2)`

∴ `4k` = 3, ∴ `k = 3/4`