What is the extreme f(x)=xln(x/3),0.1<= x<= 3 ?

The extreme f(x)=xln(x/3),0.1<= x<= 3 is Maximum(0.1,-0.34011…),Minimum(3/e ,-3/e),Maximum(3,0)

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extreme f(x)=xln(x/3),0.1<= x<= 3

extreme points

f (x) = x ln (\frac{x}{3}), 0.1 \leq x \leq 3

Maximum (0.1, - 0.34011 \dots), Minimum (\frac{3}{e}, - \frac{3}{e}), Maximum (3, 0)

Solution steps

Find the critical points:

x = 0.1, x = \frac{3}{e}, x = 3

Domain of

x ln (\frac{x}{3}), 0.1 \leq x \leq 3 : 0.1 \leq x \leq 3

Find intervals:

Decreasing

: 0.1 < x < \frac{3}{e},

Increasing

: \frac{3}{e} < x < 3

Plug the extreme point

x = 0.1

into

x ln (\frac{x}{3}) \Rightarrow y = - 0.34011 \dots

Maximum (0.1, - 0.34011 \dots)

Plug the extreme point

x = \frac{3}{e}

into

x ln (\frac{x}{3}) \Rightarrow y = - \frac{3}{e}

Minimum (\frac{3}{e}, - \frac{3}{e})

Plug the extreme point

x = 3

into

x ln (\frac{x}{3}) \Rightarrow y = 0

Maximum (3, 0)

Maximum (0.1, - 0.34011 \dots), Minimum (\frac{3}{e}, - \frac{3}{e}), Maximum (3, 0)

f (x) = 2 x^{3} - 6 x^{2} + 6 x - 8

x^{3} - 27 x + 51

x^{3} - 27 x + 57

f (x) = 2 x^{3} - 6 x^{2} + 6 x + 1

f (x) = (x + 3) - 5

What is the extreme f(x)=xln(x/3),0.1<= x<= 3 ?
The extreme f(x)=xln(x/3),0.1<= x<= 3 is Maximum(0.1,-0.34011…),Minimum(3/e ,-3/e),Maximum(3,0)