What is the inflection 2sin(x)+sin(2x) ?

The inflection 2sin(x)+sin(2x) is

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inflection 2sin(x)+sin(2x)

inflection points

2 sin (x) + sin (2 x)

(2 πn, 0), (arccos (- \frac{1}{4}) + 2 πn, \frac{15}{2} + sin (2 arccos (- \frac{1}{4}))), (π + 2 πn, 0), (- arccos (- \frac{1}{4}) + 2 π + 2 πn, - \frac{15}{2} + sin (2 (- arccos (- \frac{1}{4}) + 2 π)))

Solution steps

Find where

f^{''} (x)

is equal to zero or undefined

x = 2 πn, x = arccos (- \frac{1}{4}) + 2 πn, x = π + 2 πn, x = - arccos (- \frac{1}{4}) + 2 π + 2 πn

Identify inflection points not in domain or where

f (x)

is not continuous

Domain of

2 sin (x) + sin (2 x) : - \infty < x < \infty

All inflection points are in domain and not on domain edges

x = 2 πn, x = arccos (- \frac{1}{4}) + 2 πn, x = π + 2 πn, x = - arccos (- \frac{1}{4}) + 2 π + 2 πn

Find intervals:

Concave Downward

: 2 πn < x < arccos (- \frac{1}{4}) + 2 πn,

Concave Upward

: arccos (- \frac{1}{4}) + 2 πn < x < π + 2 πn,

Concave Downward

: π + 2 πn < x < - arccos (- \frac{1}{4}) + 2 π + 2 πn,

Concave Upward

: - arccos (- \frac{1}{4}) + 2 π + 2 πn < x < 2 π + 2 πn

Plug

x = 2 πn

into

2 sin (x) + sin (2 x) : 0

(2 πn, 0)

Plug

x = arccos (- \frac{1}{4}) + 2 πn

into

2 sin (x) + sin (2 x) : \frac{15}{2} + sin (2 arccos (- \frac{1}{4}))

(arccos (- \frac{1}{4}) + 2 πn, \frac{15}{2} + sin (2 arccos (- \frac{1}{4})))

Plug

x = π + 2 πn

into

2 sin (x) + sin (2 x) : 0

(π + 2 πn, 0)

Plug

x = - arccos (- \frac{1}{4}) + 2 π + 2 πn

into

2 sin (x) + sin (2 x) : - \frac{15}{2} + sin (2 (- arccos (- \frac{1}{4}) + 2 π))

(- arccos (- \frac{1}{4}) + 2 π + 2 πn, - \frac{15}{2} + sin (2 (- arccos (- \frac{1}{4}) + 2 π)))

(2 πn, 0), (arccos (- \frac{1}{4}) + 2 πn, \frac{15}{2} + sin (2 arccos (- \frac{1}{4}))), (π + 2 πn, 0), (- arccos (- \frac{1}{4}) + 2 π + 2 πn, - \frac{15}{2} + sin (2 (- arccos (- \frac{1}{4}) + 2 π)))

\frac{x ^{2} + 7 x + 12}{- 2 x ^{2} - 2 x + 12}

f (x) = lo g_{6} (4 x + 4)

- 8 x^{2} + 4

f (x) = - 2 x^{2} + 4 x - 7

\frac{1}{5 + e ^{3 x}}