given y = 4x-3
find the minimum value of xy.
find the rate of change of xy with respect to x.
find the rate of change of xy with respect to y.
f(x,y) = xy = x (4x-3)
f = 4 x^2 - 3x
df/dx = 8 x - 3
that is 0 when x = 3/8 and the second derivative is positive so it is a minumum
then x y = (3/8)(3/2 -3)
= (3/8) (-3/2) = -9/16
we found df/dx
now df/dy
4x = y+3
x = (y+3)/4
g(y) = xy = y(y+3)/4
= (1/4)(y^2+3y)
dg/dy = (1/4)(2 y+3)
= y/2 + 3/4
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.
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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.
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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.
To find the minimum value of xy, we can first find the critical point by taking the derivative of xy with respect to x and setting it equal to zero. Then, we can substitute the x-value of the critical point into the original equation to find the minimum value of xy.
Step 1: Find the derivative of xy with respect to x:
To take the derivative of xy, we can use the product rule. The product rule states that if we have two functions u(x) and v(x), the derivative of their product (u(x) * v(x)) is given by the following formula:
(d/dx) [u(x) * v(x)] = u(x) * dv/dx + v(x) * du/dx
In this case, our u(x) is x and v(x) is y(x) = 4x - 3. Let's differentiate the equation:
dy/dx = d/dx [4x - 3]
= 4
Step 2: Set dy/dx equal to zero and solve for x:
To find the critical points, we set the derivative equal to zero and solve for x:
4 = 0
This equation has no solution, which means there are no critical points.
Step 3: Substitute the x-value back into the original equation:
Since there are no critical points, we need to consider the endpoints of the domain to find the minimum value of xy. However, since the equation y = 4x - 3 is a straight line, it extends indefinitely in both the positive and negative directions, and there are no endpoints to consider.
Therefore, the minimum value of xy does not exist.
To find the rate of change of xy with respect to x, we can simply take the derivative of xy with respect to x.
d(xy)/dx = d/dx [xy]
= x * dy/dx + y * dx/dx (using the product rule)
= x * (dy/dx) + y
Substituting our value for dy/dx from before:
d(xy)/dx = x * (4) + y
= 4x + y
To find the rate of change of xy with respect to y, we can similarly take the derivative of xy with respect to y.
d(xy)/dy = d/dy [xy]
= x * dy/dy + y * dx/dy (using the product rule)
= x + y * dx/dy
Since y does not explicitly depend on y (it only depends on x), dx/dy is equal to zero. Therefore:
d(xy)/dy = x + y * 0
= x