Question

Consider the function f(x)=xlnx.For the following questions, write inf for ?, -inf for ??, and None if no such answer exists.

a.) The domain of this function is .
b.) The x-intercept is .
c.) f?(x) = .
d.) f(x) is increasing on the interval and decreasing on .
e.) f(x) has a local minimum at and a local maximum at .
f.) f?(x) = .
g.) f(x) is concave up on the interval and concave down on .
h.) f(x) has a point of inflection at .

Answer 1

The domain of the function is (0, +inf). The function is increasing on (0, +inf) and decreasing on (-inf, 0). It has a local minimum at x = 1 and a local maximum at x = e.

Step-by-step explanation:

a.) The domain of the function is (0, +inf). The function is defined for all positive values of x, but not including x = 0.

b.) The x-intercept is None. The function does not have an x-intercept because it is not equal to zero for any value of x.

c.) f'(x) = ln(x) + 1.

d.) f(x) is increasing on the interval (0, +inf) and decreasing on the interval (-inf, 0).

e.) f(x) has a local minimum at x = 1 and a local maximum at x = e, where e is the base of natural logarithm.

f.) f''(x) = 1/x.

g.) f(x) is concave up on the interval (0, +inf) and concave down on the interval (-inf, 0).

h.) f(x) does not have a point of inflection.