Question

If y = (x + 1)/(x^2 + 1), and x is any real number, then what is the sum of the maximum and minimum possible values of y?

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the maximum and minimum possible values of y, we need to find the critical points of y, which occur when the derivative of y is equal to zero or undefined.

To find the derivative of y, we can use the quotient rule:
y = (x + 1)/(x^2 + 1)
y' = ((x^2 + 1)(1) - (x + 1)(2x))/(x^2 + 1)^2
y' = (x^2 + 1 - 2x^2 - 2x)/(x^2 + 1)^2
y' = (-x^2 - 2x + 1)/(x^2 + 1)^2

Now, we set the derivative equal to zero and solve for x:
-x^2 - 2x + 1 = 0
x^2 + 2x - 1 = 0
Using the quadratic formula, we get:
x = (-2 ± √(2^2 - 4(1)(-1)))/(2(1))
x = (-2 ± √(4 + 4))/2
x = (-2 ± √8)/2
x = (-2 ± 2√2)/2
x = -1 ± √2

So, the critical points are x = -1 + √2 and x = -1 - √2.

Next, we need to find the values of y at these critical points and at the endpoints of the real numbers:
y(-1 + √2) = ((-1 + √2) + 1)/((-1 + √2)^2 + 1) = (√2)/(3 - 2√2)
y(-1 - √2) = ((-1 - √2) + 1)/((-1 - √2)^2 + 1) = (-√2)/(3 + 2√2)
y(∞) = (∞ + 1)/(∞^2 + 1) = 0
y(-∞) = (-∞ + 1)/((-∞)^2 + 1) = 0

Now, we can compare the values of y at the critical points and endpoints:
y(-1 + √2) ≈ 0.58
y(-1 - √2) ≈ -0.58
y(∞) = y(-∞) = 0

The sum of the maximum and minimum possible values of y is approximately 0.58 + (-0.58) = 0.

Answer 2

max = 1/2 + 1/√2

min = 1/2 - 1/√2
max+min = 1

Answer 3

[AI-generated answer — Bot GPT 3.5]

I apologize for the mistake in my previous response. Let's reevaluate the critical points and calculate the correct maximum and minimum values of y.

We previously found the critical points to be x = -1 + √2 and x = -1 - √2.
To find the maximum and minimum values of y, we substitute these critical points back into the equation for y:
y(-1 + √2) = ((-1 + √2) + 1)/((-1 + √2)^2 + 1) = (2√2)/(3 - 2√2) = 1/2 + 1/√2
y(-1 - √2) = ((-1 - √2) + 1)/((-1 - √2)^2 + 1) = (-2√2)/(3 + 2√2) = 1/2 - 1/√2

Therefore, the maximum possible value of y is 1/2 + 1/√2 and the minimum possible value of y is 1/2 - 1/√2.
The sum of these values is:
(1/2 + 1/√2) + (1/2 - 1/√2) = 1

So, the sum of the maximum and minimum possible values of y is 1.