Question

Consider the curve given by the equation (2y - 1)^3 - 24x = -3. Find the value of y'' when x = 2.

a) 2
b) 3
c) 4
d) 5

Answer 1

To find the value of y'' when x = 2, differentiate the equation twice using the chain rule and product rule. Then substitute x = 2 into the equation to get the value of y''.

Step-by-step explanation:

To find the value of y'' when x = 2, we need to find the second derivative of the given equation and substitute x = 2 into it. Let's start by differentiating the equation:

(2y - 1)^3 - 24x = -3

Using the chain rule, we get:

6(2y - 1)^2 * (2y - 1) * dy/dx - 24 = 0

Now, let's differentiate again using the product rule:

12(2y - 1) * (2y - 1) * dy/dx + 6(2y - 1)^2 * d^2y/dx^2 = 0

Substituting x = 2, we get:

12(2y - 1) * (2y - 1) * dy/dx + 6(2y - 1)^2 * d^2y/dx^2 = 0

Now, we can solve for d^2y/dx^2:

d^2y/dx^2 = -12(2y - 1) * (2y - 1) * dy/dx / (6(2y - 1)^2)

Now, substitute x = 2 into the equation:

d^2y/dx^2 = -12(2y - 1) * (2y - 1) * dy/dx / (6(2y - 1)^2)

d^2y/dx^2 = -12(2y - 1) / 6 = -2(2y - 1).

Now, substituting x = 2, we get:

y'' = -2(2y - 1) = -4y + 2

So, the value of y'' when x = 2 is -4y + 2.