What is the extreme f(x)= x/(x^2+1),0<= x<= 2 ?

The extreme f(x)= x/(x^2+1),0<= x<= 2 is Minimum(0,0),Maximum(1, 1/2),Minimum(2, 2/5)

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extreme f(x)= x/(x^2+1),0<= x<= 2

extreme points

f (x) = \frac{x}{x ^{2} + 1}, 0 \leq x \leq 2

Minimum (0, 0), Maximum (1, \frac{1}{2}), Minimum (2, \frac{2}{5})

Solution steps

Find the critical points:

x = 0, x = 1, x = 2

Domain of

\frac{x}{x ^{2} + 1}, 0 \leq x \leq 2 : 0 \leq x \leq 2

Find intervals:

Increasing

: 0 < x < 1,

Decreasing

: 1 < x < 2

Plug the extreme point

x = 0

into

\frac{x}{x ^{2} + 1} \Rightarrow y = 0

Minimum (0, 0)

Plug the extreme point

x = 1

into

\frac{x}{x ^{2} + 1} \Rightarrow y = \frac{1}{2}

Maximum (1, \frac{1}{2})

Plug the extreme point

x = 2

into

\frac{x}{x ^{2} + 1} \Rightarrow y = \frac{2}{5}

Minimum (2, \frac{2}{5})

Minimum (0, 0), Maximum (1, \frac{1}{2}), Minimum (2, \frac{2}{5})

f (x) = \frac{x ^{3}}{3} - 2 x^{2} - 12 x

10 x y

f (x, y) = 64 - x^{2} - y^{2}

f (x) = 3 x^{4} + 4 x^{3} - 12 x^{2} + 10

x 4 - x

What is the extreme f(x)= x/(x^2+1),0<= x<= 2 ?
The extreme f(x)= x/(x^2+1),0<= x<= 2 is Minimum(0,0),Maximum(1, 1/2),Minimum(2, 2/5)