Question

If f(x) is continuous at x = 3, where

f(x) = ax + 1, for x ≤ 3

= bx + 3, for x > 3 then.

विकल्प

• `"a + b" = (-2)/3`

• `"a - b" = (-2)/3`

• `"a - b" = 2/3`

• `"a + b" = 2/3`

Answer 1

`"a - b" = 2/3`

Explanation:

We have, f(x) is continuous at x = 3

where f(x) = `{("a"x + 1  "for"  x ≤ 3),("b"x + 3  "for"  x > 3):},`

`therefore lim_(x->3^-) "f"(x) = lim_(x->3^+)`f(x) = f(3)     .....(i)

Now, `lim_(x->3^-) "f"(x) = lim_("h" -> 0) "f"(3 - "h")`

`lim_("h" -> 0) "a"(3 - "h") + 1`

= 3a + 1     ....(ii)

`lim_(x->3^+) "f"(x) = lim_("h" -> 0) "f"(3 + "h")`

`lim_("h" -> 0) "b"(3 - "h") + 3`

= 3b + 3    ....(iii)

and f(3) = 3a + 1       ....(iv)

Since, f(x) is continuous at x = 3

∴ From Eqs. (i), (ii), (iii) and (iv)

we get  3a + 1 = 3b + 3

⇒ 3a - 3b = 2

⇒ a - b = `2/3`