Determine the intervals on which the function f(x)= x*lnx restricted to positive x is increasing and intervals on which the function is decreasing. Check your answers by graphing the function.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function f(x) = x*ln x. It is increasing in the interval where f'(x) > 0 and decreasing in the interval where f'(x) < 0

f'(x) = x/x + ln x = 1 + ln x

1 + ln x > 0

=> ln x > -1

=> x > 1/e

1 + ln x < 0

=> ln x < -1

=> x < 1/e

The function is increasing in the interval {1/e, inf.} and decreasing in the interval {0, 1/e}

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