Consider the following function.

f(x) = (2 − x)^(e^−x)
(a) Find the intervals of increase or decrease.

(b) Find the intervals of concavity.

(c) Find the point of inflection.
(x, y) =

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To find the intervals of increase or decrease of a function, we need to find its derivative. Using the chain rule, we can find the derivative of f(x).

(a) Derivative of f(x):
f'(x) = (-1)(2-x)^(e^-x) + (2-x)(e^-x)(-e^-x)
= -(2-x)^(e^-x) + (2-x)(e^-2x)

To determine the intervals of increase or decrease, we need to find where f'(x) is positive or negative. Here's how you can do it:

1. Find the critical points by setting f'(x) equal to zero and solving for x:
-(2-x)^(e^-x) + (2-x)(e^-2x) = 0

This equation is difficult to solve algebraically. You can use numerical methods, such as a calculator or software, to find the approximate values of x where f'(x) equals zero.

2. Divide the number line into intervals using the critical points.

3. Choose a test point within each interval and evaluate f'(x) at that test point.

4. If f'(x) is positive, then the function is increasing on that interval. If f'(x) is negative, then the function is decreasing on that interval.

By following these steps, you can determine the intervals of increase or decrease for the given function f(x).

(b) To find the intervals of concavity, we need to find the second derivative of f(x). Let's differentiate f'(x) to get the second derivative.

f''(x) = (e^-2x)(2-x) - 2e^-x - (e^-2x)(2-x) - (e^-x)(-1)
= 2e^-2x - e^-x

To determine the intervals of concavity, we need to find where f''(x) is positive or negative. Here's how you can do it:

1. Find the critical points by setting f''(x) equal to zero and solving for x:
2e^-2x - e^-x = 0

Again, this equation is difficult to solve algebraically. You can use numerical methods to find the approximate values of x where f''(x) equals zero.

2. Divide the number line into intervals using the critical points.

3. Choose a test point within each interval and evaluate f''(x) at that test point.

4. If f''(x) is positive, then the function is concave up on that interval. If f''(x) is negative, then the function is concave down on that interval.

By following these steps, you can determine the intervals of concavity for the given function f(x).

(c) To find the point of inflection, we need to find where the concavity changes. In other words, we need to find where f''(x) equals zero or is undefined. However, in this case, f''(x) is always defined for all values of x.

To find the point of inflection, we need to solve the equation f''(x) = 0 and check if there is a sign change in the intervals on either side of the critical point.

Since f''(x) = 2e^-2x - e^-x, we need to solve the equation 2e^-2x - e^-x = 0 for x. Again, this equation is difficult to solve algebraically, so you may need to use numerical methods to find the approximate values of x where f''(x) equals zero.

Once you have the critical points, you can check the sign of f''(x) on either side of each critical point to determine if there is a change in concavity. If there is a sign change, then that point is a point of inflection.

By following these steps, you can find the point of inflection for the given function f(x) and obtain the coordinates (x, y) of the point.