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

The extreme f(x)=sin(x)+cos^2(x) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

extreme f(x)=sin(x)+cos^2(x)

extreme points

f (x) = sin (x) + cos^{2} (x)

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

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

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

Plug the extreme point

x = \frac{π}{2} + 2 πn

into

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

Minimum (\frac{π}{2} + 2 πn, 1)

Plug the extreme point

x = \frac{5 π}{6} + 2 πn

into

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

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

Plug the extreme point

x = \frac{3 π}{2} + 2 πn

into

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

Minimum (\frac{3 π}{2} + 2 πn, - 1)

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

f (x) = cos (5 x)

\frac{x}{( x - 1 ) ^{2}}

f (x) = cos (4 x)

f (x, y) = x y^{2} + 2 x y

e^{- 1.5 x^{2}}