What is the extreme f(x)=2sin(x)-cos(2x),0<= x<= 2pi ?

Q: What is the extreme f(x)=2sin(x)-cos(2x),0<= x<= 2pi ?

The extreme f(x)=2sin(x)-cos(2x),0<= x<= 2pi is

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Popular >

extreme f(x)=2sin(x)-cos(2x),0<= x<= 2pi

Solution

extreme points

f (x) = 2 sin (x) - cos (2 x), 0 \leq x \leq 2 π

Solution

Minimum (0, - 1), Maximum (\frac{π}{2}, 3), Minimum (\frac{7 π}{6}, - \frac{3}{2}), Maximum (\frac{3 π}{2}, - 1), Minimum (\frac{11 π}{6}, - \frac{3}{2}), Maximum (2 π, - 1)

Solution steps

Find the critical points:

x = 0, x = \frac{π}{2}, x = \frac{7 π}{6}, x = \frac{3 π}{2}, x = \frac{11 π}{6}, x = 2 π

Domain of

2 sin (x) - cos (2 x), 0 \leq x \leq 2 π : 0 \leq x \leq 2 π

Find intervals:

Increasing

: 0 < x < \frac{π}{2},

Decreasing

: \frac{π}{2} < x < \frac{7 π}{6},

Increasing

: \frac{7 π}{6} < x < \frac{3 π}{2},

Decreasing

: \frac{3 π}{2} < x < \frac{11 π}{6},

Increasing

: \frac{11 π}{6} < x < 2 π

Plug the extreme point

x = 0

into

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

Minimum (0, - 1)

Plug the extreme point

x = \frac{π}{2}

into

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

Maximum (\frac{π}{2}, 3)

Plug the extreme point

x = \frac{7 π}{6}

into

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

Minimum (\frac{7 π}{6}, - \frac{3}{2})

Plug the extreme point

x = \frac{3 π}{2}

into

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

Maximum (\frac{3 π}{2}, - 1)

Plug the extreme point

x = \frac{11 π}{6}

into

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

Minimum (\frac{11 π}{6}, - \frac{3}{2})

Plug the extreme point

x = 2 π

into

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

Maximum (2 π, - 1)

Minimum (0, - 1), Maximum (\frac{π}{2}, 3), Minimum (\frac{7 π}{6}, - \frac{3}{2}), Maximum (\frac{3 π}{2}, - 1), Minimum (\frac{11 π}{6}, - \frac{3}{2}), Maximum (2 π, - 1)

Graph

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Frequently Asked Questions (FAQ)

What is the extreme f(x)=2sin(x)-cos(2x),0<= x<= 2pi ?
The extreme f(x)=2sin(x)-cos(2x),0<= x<= 2pi is