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

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

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

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extreme sin(2x),0<= x<= 2pi

Solution

extreme points

sin (2 x), 0 \leq x \leq 2 π

Solution

Minimum (0, 0), Maximum (\frac{π}{4}, 1), Minimum (\frac{3 π}{4}, - 1), Maximum (\frac{5 π}{4}, 1), Minimum (\frac{7 π}{4}, - 1), Maximum (2 π, 0)

Solution steps

Find the critical points:

x = 0, x = \frac{π}{4}, x = \frac{3 π}{4}, x = \frac{5 π}{4}, x = \frac{7 π}{4}, x = 2 π

Domain of

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

Find intervals:

Increasing

: 0 < x < \frac{π}{4},

Decreasing

: \frac{π}{4} < x < \frac{3 π}{4},

Increasing

: \frac{3 π}{4} < x < \frac{5 π}{4},

Decreasing

: \frac{5 π}{4} < x < \frac{7 π}{4},

Increasing

: \frac{7 π}{4} < x < 2 π

Plug the extreme point

x = 0

into

sin (2 x) \Rightarrow y = 0

Minimum (0, 0)

Plug the extreme point

x = \frac{π}{4}

into

sin (2 x) \Rightarrow y = 1

Maximum (\frac{π}{4}, 1)

Plug the extreme point

x = \frac{3 π}{4}

into

sin (2 x) \Rightarrow y = - 1

Minimum (\frac{3 π}{4}, - 1)

Plug the extreme point

x = \frac{5 π}{4}

into

sin (2 x) \Rightarrow y = 1

Maximum (\frac{5 π}{4}, 1)

Plug the extreme point

x = \frac{7 π}{4}

into

sin (2 x) \Rightarrow y = - 1

Minimum (\frac{7 π}{4}, - 1)

Plug the extreme point

x = 2 π

into

sin (2 x) \Rightarrow y = 0

Maximum (2 π, 0)

Minimum (0, 0), Maximum (\frac{π}{4}, 1), Minimum (\frac{3 π}{4}, - 1), Maximum (\frac{5 π}{4}, 1), Minimum (\frac{7 π}{4}, - 1), Maximum (2 π, 0)

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Frequently Asked Questions (FAQ)

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