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

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

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

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Popular >

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

Solution

extreme points

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

Solution

Minimum (0, 0), Maximum (\frac{π}{4}, \frac{1}{2}), Minimum (\frac{3 π}{4}, - \frac{1}{2}), Maximum (\frac{5 π}{4}, \frac{1}{2}), Minimum (\frac{7 π}{4}, - \frac{1}{2}), 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 (x) cos (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 (x) cos (x) \Rightarrow y = 0

Minimum (0, 0)

Plug the extreme point

x = \frac{π}{4}

into

sin (x) cos (x) \Rightarrow y = \frac{1}{2}

Maximum (\frac{π}{4}, \frac{1}{2})

Plug the extreme point

x = \frac{3 π}{4}

into

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

Minimum (\frac{3 π}{4}, - \frac{1}{2})

Plug the extreme point

x = \frac{5 π}{4}

into

sin (x) cos (x) \Rightarrow y = \frac{1}{2}

Maximum (\frac{5 π}{4}, \frac{1}{2})

Plug the extreme point

x = \frac{7 π}{4}

into

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

Minimum (\frac{7 π}{4}, - \frac{1}{2})

Plug the extreme point

x = 2 π

into

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

Maximum (2 π, 0)

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

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Frequently Asked Questions (FAQ)

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