Find the relationship between ‘a’ and ’b’ so that the function ‘f’ defined by:

is continuous at x = 3 .

OR

If x^y = e^{x– y}, show that

Given,

…(1)

It is given that f(x) is continuous at x = 3

By definition of continuity:

∴ LHL at (x = 3) = RHL at (x = 3) = f(3)

⇒

To relate ‘a’ and ‘b’ we can consider RHL = f(3), as calculation will be easier and fast and both the values will be related. You can take LHL = RHL also.

∴

⇒

⇒ b(3+0) + 3 = 3a + 1

⇒ 3b + 3 = 3a + 1

⇒ 3a – 3b = 2 is the required relation.

OR

Given, x^y = e^{x – y}

Taking log both sides –

⇒ y log x = (x – y) log_e e

⇒ y log x = x – y …(1)

⇒ x = y (1 + log x) = y log (xe) …(2)

∵ y log x = x - y

Differentiating w.r.t x , we get –

Applying product rule and chain rule of differentiation-

⇒

⇒

⇒

⇒ [∵ log_e e = 1]

⇒

⇒ …[from eqn 1]

⇒ …[putting the value of y/x form (2)]