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

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

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

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Popular >

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

Solution

extreme points

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

Solution

Minimum (0, - 1), Maximum (\frac{π}{2}, 1), Minimum (\frac{7 π}{6}, - \frac{5}{4}), Maximum (\frac{3 π}{2}, - 1), Minimum (\frac{11 π}{6}, - \frac{5}{4}), 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

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

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

Minimum (0, - 1)

Plug the extreme point

x = \frac{π}{2}

into

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

Maximum (\frac{π}{2}, 1)

Plug the extreme point

x = \frac{7 π}{6}

into

sin (x) - cos^{2} (x) \Rightarrow y = - \frac{5}{4}

Minimum (\frac{7 π}{6}, - \frac{5}{4})

Plug the extreme point

x = \frac{3 π}{2}

into

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

Maximum (\frac{3 π}{2}, - 1)

Plug the extreme point

x = \frac{11 π}{6}

into

sin (x) - cos^{2} (x) \Rightarrow y = - \frac{5}{4}

Minimum (\frac{11 π}{6}, - \frac{5}{4})

Plug the extreme point

x = 2 π

into

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

Maximum (2 π, - 1)

Minimum (0, - 1), Maximum (\frac{π}{2}, 1), Minimum (\frac{7 π}{6}, - \frac{5}{4}), Maximum (\frac{3 π}{2}, - 1), Minimum (\frac{11 π}{6}, - \frac{5}{4}), Maximum (2 π, - 1)

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Frequently Asked Questions (FAQ)

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