What is the inflection sin(3x) ?

The inflection sin(3x) is ((2pi)/3 n,0),(pi/3+(2pi)/3 n,0)

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inflection sin(3x)

inflection points

sin (3 x)

(\frac{2 π}{3} n, 0), (\frac{π}{3} + \frac{2 π}{3} n, 0)

Solution steps

Find where

f^{''} (x)

is equal to zero or undefined

x = \frac{2 π}{3} n, x = \frac{π}{3} + \frac{2 π}{3} n

Identify inflection points not in domain or where

f (x)

is not continuous

Domain of

sin (3 x) : - \infty < x < \infty

All inflection points are in domain and not on domain edges

x = \frac{2 π}{3} n, x = \frac{π}{3} + \frac{2 π}{3} n

Find intervals:

Concave Downward

: \frac{2 π}{3} n < x < \frac{π}{3} + \frac{2 π}{3} n,

Concave Upward

: \frac{π}{3} + \frac{2 π}{3} n < x < \frac{2 π}{3} + \frac{2 π}{3} n

Plug

x = \frac{2 π}{3} n

into

sin (3 x) : 0

(\frac{2 π}{3} n, 0)

Plug

x = \frac{π}{3} + \frac{2 π}{3} n

into

sin (3 x) : 0

(\frac{π}{3} + \frac{2 π}{3} n, 0)

(\frac{2 π}{3} n, 0), (\frac{π}{3} + \frac{2 π}{3} n, 0)

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

f (x) = \frac{2 x - 18}{3 x ^{2} - 20 x - 63}

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

(- 3, 10), (20, - 5)

x^{\frac{1}{5}} (x + 6)

What is the inflection sin(3x) ?
The inflection sin(3x) is ((2pi)/3 n,0),(pi/3+(2pi)/3 n,0)