What is the extreme f(x)=xe^{-6x},0<= x<= 2 ?

The extreme f(x)=xe^{-6x},0<= x<= 2 is Minimum(0,0),Maximum(1/6 , 1/(6e)),Minimum(2, 2/(e^{12)})

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extreme f(x)=xe^{-6x},0<= x<= 2

extreme

Minimum (0, 0), Maximum (\frac{1}{6}, \frac{1}{6 e}), Minimum (2, \frac{2}{e ^{12}})

Solution steps

Find the critical points:

x = 0, x = \frac{1}{6}, x = 2

Domain of

x e^{- 6 x}, 0 \leq x \leq 2 : 0 \leq x \leq 2

Find intervals:

Increasing

: 0 < x < \frac{1}{6},

Decreasing

: \frac{1}{6} < x < 2

Plug the extreme point

x = 0

into

x e^{- 6 x} \Rightarrow y = 0

Minimum (0, 0)

Plug the extreme point

x = \frac{1}{6}

into

x e^{- 6 x} \Rightarrow y = \frac{1}{6 e}

Maximum (\frac{1}{6}, \frac{1}{6 e})

Plug the extreme point

x = 2

into

x e^{- 6 x} \Rightarrow y = \frac{2}{e ^{12}}

Minimum (2, \frac{2}{e ^{12}})

Minimum (0, 0), Maximum (\frac{1}{6}, \frac{1}{6 e}), Minimum (2, \frac{2}{e ^{12}})

What is the extreme f(x)=xe^{-6x},0<= x<= 2 ?
The extreme f(x)=xe^{-6x},0<= x<= 2 is Minimum(0,0),Maximum(1/6 , 1/(6e)),Minimum(2, 2/(e^{12)})