What is the inflection f(x)=x+sin(x) ?

The inflection f(x)=x+sin(x) is (2pin,2pin+sin(2pin)),(pi+2pin,pi+2pin+sin(pi+2pin))

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inflection f(x)=x+sin(x)

inflection points

f (x) = x + sin (x)

(2 πn, 2 πn + sin (2 πn)), (π + 2 πn, π + 2 πn + sin (π + 2 πn))

Solution steps

Find where

f^{''} (x)

is equal to zero or undefined

x = 2 πn, x = π + 2 πn

Identify inflection points not in domain or where

f (x)

is not continuous

Domain of

x + sin (x) : - \infty < x < \infty

All inflection points are in domain and not on domain edges

x = 2 πn, x = π + 2 πn

Find intervals:

Undefined

: E m pt y I n t er v a l,

Concave Downward

: 2 πn < x < π + 2 πn,

Concave Upward

: π + 2 πn < x < 2 π + 2 πn

Plug

x = 2 πn

into

x + sin (x) : 2 πn + sin (2 πn)

(2 πn, 2 πn + sin (2 πn))

Plug

x = π + 2 πn

into

x + sin (x) : π + 2 πn + sin (π + 2 πn)

(π + 2 πn, π + 2 πn + sin (π + 2 πn))

(2 πn, 2 πn + sin (2 πn)), (π + 2 πn, π + 2 πn + sin (π + 2 πn))

3 x + y = 3

f (x) = 7 x - 9

6 x^{4} - x^{3} + 16 x^{2} - 12

3 x

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

What is the inflection f(x)=x+sin(x) ?
The inflection f(x)=x+sin(x) is (2pin,2pin+sin(2pin)),(pi+2pin,pi+2pin+sin(pi+2pin))