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Popular >

inflection f(x)=x+cos(2x)

Solution

inflection points

f (x) = x + cos (2 x)

Solution

(\frac{π}{4} + πn, \frac{π}{4} + πn + cos (2 (\frac{π}{4} + πn))), (\frac{3 π}{4} + πn, \frac{3 π}{4} + πn + cos (2 (\frac{3 π}{4} + πn)))

Solution steps

Find where

f^{''} (x)

is equal to zero or undefined

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Identify inflection points not in domain or where

f (x)

is not continuous

Domain of

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

All inflection points are in domain and not on domain edges

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Find intervals:

Concave Downward

: πn \leq x < \frac{π}{4} + πn,

Concave Upward

: \frac{π}{4} + πn < x < \frac{3 π}{4} + πn,

Concave Downward

: \frac{3 π}{4} + πn < x < π + πn

Plug

x = \frac{π}{4} + πn

into

x + cos (2 x) : \frac{π}{4} + πn + cos (2 (\frac{π}{4} + πn))

(\frac{π}{4} + πn, \frac{π}{4} + πn + cos (2 (\frac{π}{4} + πn)))

Plug

x = \frac{3 π}{4} + πn

into

x + cos (2 x) : \frac{3 π}{4} + πn + cos (2 (\frac{3 π}{4} + πn))

(\frac{3 π}{4} + πn, \frac{3 π}{4} + πn + cos (2 (\frac{3 π}{4} + πn)))

(\frac{π}{4} + πn, \frac{π}{4} + πn + cos (2 (\frac{π}{4} + πn))), (\frac{3 π}{4} + πn, \frac{3 π}{4} + πn + cos (2 (\frac{3 π}{4} + πn)))

Popular Examples

range of x^2+6x+3 range

x^{2} + 6 x + 3

perpendicular y=-1/3 x-2,(1,3)perpendicular

y = - \frac{1}{3} x - 2, (1, 3)

line-4x+6y+1=0 line

- 4 x + 6 y + 1 = 0

extreme f(x)= 8/(x^2+4)extreme points

f (x) = \frac{8}{x ^{2} + 4}

intercepts of f(x)=-2x^2+8x intercepts

f (x) = - 2 x^{2} + 8 x