Consider the differential equation dy/dx = -1 + (y^2/ x).
Let y = g(x) be the particular solution to the differential equation dy/ dx = -1 + (y^2/ x) with initial condition g(4) = 2. Does g have a relative minimum, a relative maximum, or neither at ? Justify your answer.
Well, let's put on our math clown shoes and solve this one!
To determine if g(x) has a relative minimum, maximum, or neither, we need to analyze the behavior around the point g(4) = 2. One way to do that is by finding the derivative of g(x) and looking at its sign.
Let's differentiate the given equation with respect to x:
dy/dx = -1 + (y^2 / x)
Applying the product rule and chain rule, we have:
d^2y/dx^2 = (2y dy/dx / x) - (y^2 / x^2) + (y^2 / x)
Now, let's substitute g(x) back into the equation:
d^2g(x)/dx^2 = (2g(x) dg(x)/dx / x) - (g(x)^2 / x^2) + (g(x)^2 / x)
At the point x = 4, we can plug in the initial condition g(4) = 2:
d^2g(4)/dx^2 = (2 * 2 * dg(4)/dx / 4) - (2^2 / 4^2) + (2^2 / 4)
Simplifying this, we get:
d^2g(4)/dx^2 = (4 dg(4)/dx / 4) - (4 / 16) + 1
d^2g(4)/dx^2 = dg(4)/dx - (1/4) + 1
Notice that dg(4)/dx represents the derivative of g(x) at x = 4, which is crucial for our analysis. However, we don't know the value of this derivative based on the given information. So, we can't determine the sign of d^2g(4)/dx^2, which means we can't definitively determine if g(x) has a relative minimum, maximum, or neither at x = 4.
In other words, the behavior of g(x) around x = 4 cannot be determined solely from the given initial condition and equation. It's like walking on a tightrope blindfolded while juggling rubber chickens – quite the risk!
To determine whether the function g(x) has a relative minimum, relative maximum, or neither, we need to analyze the behavior of its derivative, g'(x).
First, let's find g'(x) by differentiating the given differential equation:
dy/dx = -1 + (y^2 / x)
Using the chain rule, we can rewrite this as follows:
g'(x) = -1 + (g(x)^2 / x)
To find the critical points where g'(x) may change sign (and therefore the possibility of a relative minimum or maximum), we set g'(x) equal to zero and solve for x:
-1 + (g(x)^2 / x) = 0
Rearranging the equation, we have:
g(x)^2 = x
Taking the square root of both sides, we get:
|g(x)| = √x
Now, let's consider the initial condition given: g(4) = 2. Plugging this into the equation, we have:
|2| = √4
This simplifies to:
2 = 2
So, g(4) = 2 satisfies the equation. However, since the square root function has both positive and negative values, g(x) can take on two different values for a given x. So, g(x) is not uniquely determined, and we cannot conclude whether it has a relative minimum, relative maximum, or neither at x = 4.
In summary, due to the ambiguity caused by the absolute value in the equation |g(x)| = √x, we cannot determine whether g(x) has a relative minimum, relative maximum, or neither at x = 4.
To determine whether the particular solution g(x) has a relative minimum, relative maximum, or neither at x = 4, we need to analyze the behavior of g(x) around that point.
1. Let's rewrite the given differential equation as follows:
dy/dx = -1 + (y^2 / x)
dy/dx + 1 = y^2 / x
2. Now, we'll substitute g(x) into the differential equation and simplify:
g'(x) + 1 = g(x)^2 / x
3. We can rewrite the equation in terms of g(x) and its derivative:
g'(x) = (g(x)^2 / x) - 1
4. Since we have the initial condition g(4) = 2, we can utilize this information to determine the behavior of g(x) near x = 4. Let's examine a small interval around x = 4, such as (3.9, 4.1).
5. We can evaluate g'(x) within the interval (3.9, 4.1):
g'(x) = (g(4)^2 / 4) - 1
= (2^2 / 4) - 1
= 1 - 1
= 0
6. As g'(x) = 0 within the interval (3.9, 4.1), we can conclude that g(x) has a critical point at x = 4.
7. Now, let's investigate the behavior of g(x) on both sides of x = 4. We can consider two cases:
Case 1: x < 4
If x < 4, the equation g'(x) = (g(x)^2 / x) - 1 implies g'(x) < 0. This suggests that g(x) is decreasing on the left side of x = 4.
Case 2: x > 4
If x > 4, the equation g'(x) = (g(x)^2 / x) - 1 implies g'(x) > 0. This suggests that g(x) is increasing on the right side of x = 4.
8. Combining the information from the two cases, we can conclude that g(x) has neither a relative minimum nor a relative maximum at x = 4. The behavior of g(x) changes from decreasing to increasing at x = 4, indicating a point of inflection.
Therefore, g(x) has neither a relative minimum nor a relative maximum at x = 4.
Well where is dy/dx = 0 ? (horizontal either max or min or inflection)
dy/dx = -1 + (y^2/ x) = 0
or y^2 = x
It said start at (2, 4)
well that very spot is horizontal 4 = 2^2
now is it a min or a max there?
what is d^2y/dx^2 ? at (2,4) ?
dy/dx = -1 + (y^2/ x)
d/dx(dy/x) = 0 + d/dx(y^2/x)= [x(-2y dy/dx) -y^2] /x^2
at (2,4)= [ 2*-8* (0 at that point) - 16 ] / 4 = -32/4 oh that is a maximum because headed down