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

The extreme f(x)=2cos(x)+sin(2x) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

extreme points

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

Maximum (\frac{π}{6} + 2 πn, 3 + \frac{3}{2}), Minimum (\frac{5 π}{6} + 2 πn, - 3 - \frac{3}{2}), Saddle (\frac{3 π}{2} + 2 πn, 0)

Solution steps

Find the critical points:

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

Domain of

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

Find intervals:

Increasing

: 2 πn \leq x < \frac{π}{6} + 2 πn,

Decreasing

: \frac{π}{6} + 2 πn < x < \frac{5 π}{6} + 2 πn,

Increasing

: \frac{5 π}{6} + 2 πn < x < \frac{3 π}{2} + 2 πn,

Increasing

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

Plug the extreme point

x = \frac{π}{6} + 2 πn

into

2 cos (x) + sin (2 x) \Rightarrow y = 3 + \frac{3}{2}

Maximum (\frac{π}{6} + 2 πn, 3 + \frac{3}{2})

Plug the extreme point

x = \frac{5 π}{6} + 2 πn

into

2 cos (x) + sin (2 x) \Rightarrow y = - 3 - \frac{3}{2}

Minimum (\frac{5 π}{6} + 2 πn, - 3 - \frac{3}{2})

Plug the extreme point

x = \frac{3 π}{2} + 2 πn

into

2 cos (x) + sin (2 x) \Rightarrow y = 0

Saddle (\frac{3 π}{2} + 2 πn, 0)

Maximum (\frac{π}{6} + 2 πn, 3 + \frac{3}{2}), Minimum (\frac{5 π}{6} + 2 πn, - 3 - \frac{3}{2}), Saddle (\frac{3 π}{2} + 2 πn, 0)

f (x, y) = x^{2} + 2 y^{2}

f (x) = - x^{2} + 10 x - 21

f (x, y) = x^{3} + 2 y^{3} - 3 y^{2} - 3 x

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

f (x, y) = \frac{x}{x ^{2} + y ^{2}}