#### To determine

**To find**:

a) Intervals of increase or decrease,

b) The intervals of concavity,

c) The points of inflection.

#### Answer

(a) The interval of increase is (2,∞) and the interval of decrease is (-∞,2).

(b) f is concave upward on (-∞,0) and (0,∞).

(c) there is no inflection point

#### Explanation

**Given**: fx=exx2.

a) f'x=ex-2x-3+x-2ex=ex(-2x-3+x-2). Now, f'x>0 if exx-2-2x-1+1>0, i.e., 1-2x>0, i.e., x-2>0, i.e.,x>2. Also, f'x<0 if x<2. Therefore, the interval of increase is (2,∞) and the interval of decrease is (-∞,2).

b) f''x=6x-4ex-4 x-3ex+exx-2=exx-2(6x-2-4x-1+1). Now, f''x>0 if 6x-2-4x-1+1>0, i.e., x2-4x+6>0, i.e., x2-4x+4+2>0, i.e.,x-22+2>0. Therefore f is concave upward on (-∞,0) and (0,∞).

c) Since f''x=0 implies x-22=-2 is not possible, there is no inflection point.

**Conclusion:**

(a) the interval of increase is (2,∞) and the interval of decrease is (-∞,2).

(b) f is concave upward on (-∞,0) and (0,∞).

(c) there is no inflection point