For the function f(x)= (√1−x) / ( √1+x)

find:
a. the regions of increase and decrease,
b. the regions of concave up/down and where f(x)
has an inflection point.
c. The "regions" should be open intervals or the empty set ϕ

To find the regions of increase and decrease, we need to determine where the derivative of the function is positive or negative.

a) Regions of Increase and Decrease:
1. Differentiate the function f(x) with respect to x using the quotient rule. The derivative of f(x) is given by:

f'(x) = [(√1+x)(d(√1-x)/dx) - (√1-x)(d(√1+x)/dx)] / (1+x)^2

2. Simplify the derivative:

f'(x) = [(√1+x)(-1/2(1-x)^(-1/2)) - (√1-x)(1/2(1+x)^(-1/2))] / (1+x)^2
= -[(1+x)/√(1-x) + (1-x)/√(1+x)] / (1+x)^2

3. Now, set f'(x) equal to zero and find the critical points by solving the equation:

-(1+x)/√(1-x) - (1-x)/√(1+x) = 0

4. Solve the equation above for x. Since this equation may not have any explicit solutions, you may need to use numerical methods or graphing techniques to approximate the critical points. Let's denote these critical points as c1, c2, ...

5. Using the critical points, create intervals between them. Test a value from each interval into the derivative f'(x) to determine whether the interval is increasing or decreasing.

- If f'(x) > 0, then the function is increasing on that interval.
- If f'(x) < 0, then the function is decreasing on that interval.

The resulting intervals will be the regions of increase and decrease for f(x).

b) Regions of Concave Up/Down and Inflection Points:
1. Take the second derivative of f(x) by differentiating f'(x) with respect to x:

f''(x) = [d/dx(-[(1+x)/√(1-x) + (1-x)/√(1+x)])] / (1+x)^2

2. Simplify the second derivative:

f''(x) = [(-d(1+x)/dx√(1-x) - √(1-x)d(1+x)/dx) - (-d(1-x)/dx√(1+x) + √(1+x)d(1-x)/dx)] / (1+x)^2
= -[1/√(1-x) - (1+x)/2(1-x)^(3/2) + 1/√(1+x) + (1-x)/2(1+x)^(3/2)] / (1+x)^2
= -[2(1+x) - (1+x)√(1-x) - 2(1-x) + (1-x)√(1+x)] / (2(1-x)(1+x)^(3/2))

3. Set f''(x) equal to zero and solve for the critical points. Again, this may require numerical or graphical methods.

4. Test intervals between critical points for the sign of f''(x) to determine where the function is concave up or concave down, and if there are any inflection points.

- If f''(x) > 0, then the function is concave up on that interval.
- If f''(x) < 0, then the function is concave down on that interval.

The intervals where the concavity changes or where f''(x) equals zero represent the regions of concave up/down and the inflection points of f(x).

Note: Remember to always check the endpoints of the domain, which may also affect the conclusions about the regions, concavity, and inflection points.