Solution assignment 09 Optimizing f(x)

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Assignment 9

Verify whether the function:

y=x\ln(x)

has a maximum or a minimum.

Solution

We calculate the derivative (product rule) and make it equal to 0:

y'=1\cdot\ln(x)+x\cdot\displaystyle\frac{1}{x}=\ln(x)+1=0

This equation has the solution:

x=\displaystyle\frac{1}{e}=e^{-1}

In order to verify whether we have a maximum or a minimum, we calculate the second derivative:

y''=\displaystyle\frac{1}{x}=e>0

For x=e^{-1} there is a minimum with the coordinates: (e^{-1},-e^{-1})

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