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

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

extreme points

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

Maximum (\frac{π}{6} + 2 πn, \frac{3}{2}), Minimum (\frac{π}{2} + 2 πn, 1), Maximum (\frac{5 π}{6} + 2 πn, \frac{3}{2}), Minimum (\frac{3 π}{2} + 2 πn, - 3)

Solution steps

Find the critical points:

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

Domain of

2 sin (x) + cos (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{π}{2} + 2 πn,

Increasing

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

Decreasing

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

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

Plug the extreme point

x = \frac{π}{2} + 2 πn

into

2 sin (x) + cos (2 x) \Rightarrow y = 1

Minimum (\frac{π}{2} + 2 πn, 1)

Plug the extreme point

x = \frac{5 π}{6} + 2 πn

into

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

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

Plug the extreme point

x = \frac{3 π}{2} + 2 πn

into

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

Minimum (\frac{3 π}{2} + 2 πn, - 3)

Maximum (\frac{π}{6} + 2 πn, \frac{3}{2}), Minimum (\frac{π}{2} + 2 πn, 1), Maximum (\frac{5 π}{6} + 2 πn, \frac{3}{2}), Minimum (\frac{3 π}{2} + 2 πn, - 3)

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

f (x) = 2 - 5 x + x^{2}

f (x, y) = 33 x + 66 y + x y - x^{2} - 3 y^{2}

f (x) = 3 x + 4 32.5 - x^{2}

8 x^{2} - 8 x^{4}