What is the extreme f(x)=x+sin(x),0<= x<= 2pi ?

The extreme f(x)=x+sin(x),0<= x<= 2pi is Minimum(0,0),Saddle(pi,pi),Maximum(2pi,2pi)

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extreme f(x)=x+sin(x),0<= x<= 2pi

extreme points

f (x) = x + sin (x), 0 \leq x \leq 2 π

Minimum (0, 0), Saddle (π, π), Maximum (2 π, 2 π)

Solution steps

Find the critical points:

x = 0, x = π, x = 2 π

Domain of

x + sin (x), 0 \leq x \leq 2 π : 0 \leq x \leq 2 π

Find intervals:

Increasing

: 0 < x < π,

Increasing

: π < x < 2 π

Plug the extreme point

x = 0

into

x + sin (x) \Rightarrow y = 0

Minimum (0, 0)

Plug the extreme point

x = π

into

x + sin (x) \Rightarrow y = π

Saddle (π, π)

Plug the extreme point

x = 2 π

into

x + sin (x) \Rightarrow y = 2 π

Maximum (2 π, 2 π)

Minimum (0, 0), Saddle (π, π), Maximum (2 π, 2 π)

f (x, y) = 1 - \frac{x ^{2}}{4} - y^{2}

f (x) = x^{3} + 2

f (x) = x^{3} - 1

f (x, y) = x^{4} - \frac{18 x}{12 y ^{2}} - x y^{3} - x^{2}

y = x^{2} - z^{2}

What is the extreme f(x)=x+sin(x),0<= x<= 2pi ?
The extreme f(x)=x+sin(x),0<= x<= 2pi is Minimum(0,0),Saddle(pi,pi),Maximum(2pi,2pi)