Question

If f(x) = x² + α for x > 0 = `2sqrt(x^2 + 1)` + β for x < 0 is continuous at x = 0 and `f(1/2)` = 2 then α² + β² is ______.

Options

• 3

• `8/25`

• `25/8`

• `1/3`

Answer 1

If f(x) = x² + α for x > 0 = `2sqrt(x^2 + 1)` + β for x < 0 is continuous at x = 0 and `f(1/2)` = 2 then α² + β² is `underlinebb(25/8)`.

Explanation:

f(x) = `{{:(x^2 + α, "If"  x ≥ 0),(2sqrt(x^2 + 1) + β, "If"  x < 0):}}`

is continuous at x = 0

∴ f(0) = `lim_(x rightarrow 0^-)`f(x)

`\implies` 0 + α = `lim_(x rightarrow 0^-) 2sqrt(x^2 + 1) + β`

`\implies` α = 2 + β

`\implies` α – β = 2

`f(1/2) = (1/2)^2 + α` = 2

`\implies` α = `2 - 1/4 = 7/4`

∴ β = `7/4 - 2 = (-1)/4`

∴ α² + β² = `(7/4)^2 + ((-1)/4)^2`

= `(49 + 1)/16`

= `50/16`

= `25/8`