en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Functions >

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

Solution

extreme points

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

Solution

Maximum (\frac{π}{12} + πn, \frac{π}{12} + πn + cos (2 (\frac{π}{12} + πn))), Minimum (\frac{5 π}{12} + πn, \frac{5 π}{12} + πn + cos (2 (\frac{5 π}{12} + πn)))

Solution steps

Find the critical points:

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

Domain of

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

Find intervals:

Increasing

: πn \leq x < \frac{π}{12} + πn,

Decreasing

: \frac{π}{12} + πn < x < \frac{5 π}{12} + πn,

Increasing

: \frac{5 π}{12} + πn < x < π + πn

Plug the extreme point

x = \frac{π}{12} + πn

into

x + cos (2 x) \Rightarrow y = \frac{π}{12} + πn + cos (2 (\frac{π}{12} + πn))

Maximum (\frac{π}{12} + πn, \frac{π}{12} + πn + cos (2 (\frac{π}{12} + πn)))

Plug the extreme point

x = \frac{5 π}{12} + πn

into

x + cos (2 x) \Rightarrow y = \frac{5 π}{12} + πn + cos (2 (\frac{5 π}{12} + πn))

Minimum (\frac{5 π}{12} + πn, \frac{5 π}{12} + πn + cos (2 (\frac{5 π}{12} + πn)))

Maximum (\frac{π}{12} + πn, \frac{π}{12} + πn + cos (2 (\frac{π}{12} + πn))), Minimum (\frac{5 π}{12} + πn, \frac{5 π}{12} + πn + cos (2 (\frac{5 π}{12} + πn)))

Graph

Popular Examples

extreme f(x)=4x^3-3x^2-6x+3,0<= x<= 10

extreme f(x,y)=4x^2-3xy

extreme f(x)=xsqrt(7-x)

extreme y=5t-2u(t-1)+3u(t-5)

extreme f(x)=(x^2-x+1)/(x-1)

Frequently Asked Questions (FAQ)

What is the extreme f(x)=x+cos(2x) ?
The extreme f(x)=x+cos(2x) is