The minimum value of the function f(x)=x log x is

Options

A. `-(1)/(e)`
B. `-e`
C. `(1)/(e)`
D. `e`

Answer

A

Explanation

We have,
`f(x)=xlogx`
`f'(x)=1+logx`
For maxima or minima, put f'(x)=0
`therefore1+logx=0`
`implieslogx=-1`
`impliesx=e^(-1)`
Now, `f''(x)=(1)/(x)`
`impliesf''(e^(-1))=(1)/(e^(-1))=e gt 0`
`thereforef(x)` is decreasing.
Hence, minimum value of f(x) at `x=e^(-1)` is
`f(e^(-1))=e^(-1)loge^(-1)=(1)/(e)(-loge)=-(1)/(e)`