To find the minimum value of xy, we need to determine the critical point.
1. Differentiate the function xy with respect to x:
Using the product rule, we have:
d(xy)/dx = x(dy/dx) + y(dx/dx)
= x(dy/dx) + y
= x(d(4x-3)/dx) + y
= x(4) + y
= 4x + y
2. To find the critical point, set the derivative equal to zero:
4x + y = 0
3. Substitute the y-value from the given equation into the equation obtained in step 2:
4x + (4x - 3) = 0
8x - 3 = 0
8x = 3
x = 3/8
4. Substitute the x-value back into the given equation to find y:
y = 4(3/8) - 3
y = 3/2 - 3
y = -3/2
Therefore, the critical point is (x, y) = (3/8, -3/2). To find the minimum value of xy, substitute these values back into the equation:
xy = (3/8)(-3/2)
= -9/16
Hence, the minimum value of xy is -9/16.
---------------------------------------------------------------------
To find the rate of change of xy with respect to x, we need to differentiate the equation xy with respect to x.
Using the product rule, we have:
d(xy)/dx = x(dy/dx) + y(dx/dx)
= x(d(4x-3)/dx) + y
= x(4) + y
= 4x + y
Therefore, the rate of change of xy with respect to x is 4x + y.
---------------------------------------------------------------------
To find the rate of change of xy with respect to y, we need to differentiate the equation xy with respect to y.
Using the product rule, we have:
d(xy)/dy = x(dy/dy) + y(dx/dy)
= x + y(dx/dy)
= x + y(0)
= x
Therefore, the rate of change of xy with respect to y is x.