Question

If y = y(x) is an implicit function of x such that log_e(x + y) = 4xy, then `(d^2y)/(dx^2)` at x = 0 is equal to ______.

Options

• 20

• 30

• 40

• 50

Answer 1

If y = y(x) is an implicit function of x such that log_e(x + y) = 4xy, then `(d^2y)/(dx^2)` at x = 0 is equal to 40.

Explanation:

When x = 0 then y = 1

In (x + y) = 4xy

∴ x + y = e^4xy

Now on differentiating

⇒ 1 + y' = e^4xy(4y + 4xy')  ...(i)

At (0, 1) ⇒ y'(0) + 1 = 4 ⇒ y'(0) = 3

Now, again on differentiating equation (i)

⇒ y" = e^4xy(4y + 4xy')² + e^4xy(4y' + 4y' + 4xy")

Now, `y^('')|_(at(0,1))`

⇒ y"(0) = 1(4 × 1 + 0)² + 1(4 × 3 + 4 × 3 + 0)

⇒ y"(0) = 16 + 24 = 40

⇒ y"(0) = 40