question regarding finding if f is increasing/decreasing!?

Question

if f(x) =xlnx, then at x=1:

Which one is correct?

a) f is increasing
b) f is decreasing
c) f has a relative minimum
d) f has a relative maximum
e) f has a pt. of infelction

Answer 1

You need to evaluate the derivative at x=1.
The derivative of xln(x) is x*(1/x) + 1*ln(x) = 1 + ln(x).
When x=1, 1 + ln(x) = 1 + ln(1) + 1 + 0 = 1.
Since the derivative is positive when x = 1 the function is increasing.

Answer 2

If f (x) is positive, then f(x) is increasing. If f '(x) is negative , then f(x) is decreasing.

f'(x) = x*1/x + lnx= 1+ln(x)
f '(1) = 1+ ln(1) = 1
Hence f(x) is increasing .