What is the extreme f(x)=sqrt(3)cos(5x)+sin(5x) ?

The extreme f(x)=sqrt(3)cos(5x)+sin(5x) is

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Popular >

extreme f(x)=sqrt(3)cos(5x)+sin(5x)

Solution

extreme points

f (x) = 3 cos (5 x) + sin (5 x)

Solution

Maximum (\frac{π}{30} + 2 πn, 2), Minimum (\frac{7 π}{30} + 2 πn, - 2), Maximum (\frac{13 π}{30} + 2 πn, 2), Minimum (\frac{19 π}{30} + 2 πn, - 2), Maximum (\frac{5 π}{6} + 2 πn, 2), Minimum (\frac{31 π}{30} + 2 πn, - 2), Maximum (\frac{37 π}{30} + 2 πn, 2), Minimum (\frac{43 π}{30} + 2 πn, - 2), Maximum (\frac{49 π}{30} + 2 πn, 2), Minimum (\frac{11 π}{6} + 2 πn, - 2)

Solution steps

Find the critical points:

x = \frac{π}{30} + 2 πn, x = \frac{7 π}{30} + 2 πn, x = \frac{13 π}{30} + 2 πn, x = \frac{19 π}{30} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{31 π}{30} + 2 πn, x = \frac{37 π}{30} + 2 πn, x = \frac{43 π}{30} + 2 πn, x = \frac{49 π}{30} + 2 πn, x = \frac{11 π}{6} + 2 πn

Domain of

3 cos (5 x) + sin (5 x) : - \infty < x < \infty

Find intervals:

Increasing

: 2 πn \leq x < \frac{π}{30} + 2 πn,

Decreasing

: \frac{π}{30} + 2 πn < x < \frac{7 π}{30} + 2 πn,

Increasing

: \frac{7 π}{30} + 2 πn < x < \frac{13 π}{30} + 2 πn,

Decreasing

: \frac{13 π}{30} + 2 πn < x < \frac{19 π}{30} + 2 πn,

Increasing

: \frac{19 π}{30} + 2 πn < x < \frac{5 π}{6} + 2 πn,

Decreasing

: \frac{5 π}{6} + 2 πn < x < \frac{31 π}{30} + 2 πn,

Increasing

: \frac{31 π}{30} + 2 πn < x < \frac{37 π}{30} + 2 πn,

Decreasing

: \frac{37 π}{30} + 2 πn < x < \frac{43 π}{30} + 2 πn,

Increasing

: \frac{43 π}{30} + 2 πn < x < \frac{49 π}{30} + 2 πn,

Decreasing

: \frac{49 π}{30} + 2 πn < x < \frac{11 π}{6} + 2 πn,

Increasing

: \frac{11 π}{6} + 2 πn < x < 2 π + 2 πn

Plug the extreme point

x = \frac{π}{30} + 2 πn

into

3 cos (5 x) + sin (5 x) \Rightarrow y = 2

Maximum (\frac{π}{30} + 2 πn, 2)

Plug the extreme point

x = \frac{7 π}{30} + 2 πn

into

3 cos (5 x) + sin (5 x) \Rightarrow y = - 2

Minimum (\frac{7 π}{30} + 2 πn, - 2)

Plug the extreme point

x = \frac{13 π}{30} + 2 πn

into

3 cos (5 x) + sin (5 x) \Rightarrow y = 2

Maximum (\frac{13 π}{30} + 2 πn, 2)

Plug the extreme point

x = \frac{19 π}{30} + 2 πn

into

3 cos (5 x) + sin (5 x) \Rightarrow y = - 2

Minimum (\frac{19 π}{30} + 2 πn, - 2)

Plug the extreme point

x = \frac{5 π}{6} + 2 πn

into

3 cos (5 x) + sin (5 x) \Rightarrow y = 2

Maximum (\frac{5 π}{6} + 2 πn, 2)

Plug the extreme point

x = \frac{31 π}{30} + 2 πn

into

3 cos (5 x) + sin (5 x) \Rightarrow y = - 2

Minimum (\frac{31 π}{30} + 2 πn, - 2)

Plug the extreme point

x = \frac{37 π}{30} + 2 πn

into

3 cos (5 x) + sin (5 x) \Rightarrow y = 2

Maximum (\frac{37 π}{30} + 2 πn, 2)

Plug the extreme point

x = \frac{43 π}{30} + 2 πn

into

3 cos (5 x) + sin (5 x) \Rightarrow y = - 2

Minimum (\frac{43 π}{30} + 2 πn, - 2)

Plug the extreme point

x = \frac{49 π}{30} + 2 πn

into

3 cos (5 x) + sin (5 x) \Rightarrow y = 2

Maximum (\frac{49 π}{30} + 2 πn, 2)

Plug the extreme point

x = \frac{11 π}{6} + 2 πn

into

3 cos (5 x) + sin (5 x) \Rightarrow y = - 2

Minimum (\frac{11 π}{6} + 2 πn, - 2)

Maximum (\frac{π}{30} + 2 πn, 2), Minimum (\frac{7 π}{30} + 2 πn, - 2), Maximum (\frac{13 π}{30} + 2 πn, 2), Minimum (\frac{19 π}{30} + 2 πn, - 2), Maximum (\frac{5 π}{6} + 2 πn, 2), Minimum (\frac{31 π}{30} + 2 πn, - 2), Maximum (\frac{37 π}{30} + 2 πn, 2), Minimum (\frac{43 π}{30} + 2 πn, - 2), Maximum (\frac{49 π}{30} + 2 πn, 2), Minimum (\frac{11 π}{6} + 2 πn, - 2)

Graph

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Frequently Asked Questions (FAQ)

What is the extreme f(x)=sqrt(3)cos(5x)+sin(5x) ?
The extreme f(x)=sqrt(3)cos(5x)+sin(5x) is