What is the extreme f(x)=sin(x)-sin^2(x) ?

The extreme f(x)=sin(x)-sin^2(x) is

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extreme f(x)=sin(x)-sin^2(x)

extreme points

f (x) = sin (x) - sin^{2} (x)

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

Solution steps

Find the critical points:

x = \frac{π}{6} + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

Domain of

sin (x) - sin^{2} (x) : - \infty < x < \infty

Find intervals:

Increasing

: 2 πn \leq x < \frac{π}{6} + 2 πn,

Decreasing

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

Increasing

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

Decreasing

: \frac{5 π}{6} + 2 πn < x < \frac{3 π}{2} + 2 πn,

Increasing

: \frac{3 π}{2} + 2 πn < x < 2 π + 2 πn

Plug the extreme point

x = \frac{π}{6} + 2 πn

into

sin (x) - sin^{2} (x) \Rightarrow y = \frac{1}{4}

Maximum (\frac{π}{6} + 2 πn, \frac{1}{4})

Plug the extreme point

x = \frac{π}{2} + 2 πn

into

sin (x) - sin^{2} (x) \Rightarrow y = 0

Minimum (\frac{π}{2} + 2 πn, 0)

Plug the extreme point

x = \frac{5 π}{6} + 2 πn

into

sin (x) - sin^{2} (x) \Rightarrow y = \frac{1}{4}

Maximum (\frac{5 π}{6} + 2 πn, \frac{1}{4})

Plug the extreme point

x = \frac{3 π}{2} + 2 πn

into

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

Minimum (\frac{3 π}{2} + 2 πn, - 2)

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

x^{3} + 3 x^{2} + 5

f (x) = - \frac{4}{3} x^{3} - \frac{21}{2} x^{2} - 5 x + 12

\frac{x + 2}{x ^{2} - 3 x - 10}

y = x^{2} + 10 x + 15

f (x) = 2 - \frac{50}{x ^{2}}