Question

Find (a) the intervals of increase or decrease, (b) the intervals of concavity, and (c) the points of inflection.

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

Answer 1

Step 1:

The function is .

Differentiate with respect to .

Again differentiate with respect to .

Step 2:

To find the intervals of increase and decrease, set .

cannot be equated to zero.

Therefore intervals are and

Consider a test point from the interval .

Let in .

then is decreasing in the interval .

Consider a test point from the interval

Let in .

then is increasing in the interval .

Therefore,

is decreasing in the interval and increasing in the interval .

Answer 2

Contd..

Step 3:

To find the inflection points, set .

cannot be equated to zero.

Substitute in

Inflection point is .

Step 4:

 The test intervals are and .

Interval     Test Value              Sign of f''(x)                                 Conclusion

                   Concave upward

                   Concave downward

Solutions:

(a) is decreasing in the interval and increasing in the interval .

(b) Inflection point is .

(c) is concave up in the interval and concave down in the interval