Ekstremne vrednosti funkcije - formule i zadaci -

Ekstremne vrednosti funkcije- formule i zadaci -

2010/2011

(Ekstremi funkcije - formule i zadaci) 2010/2011 1 / 1

Monotonost i ekstremne vrednosti funkcije

Funkcija f (x) je rastuca na intervalu (a, b) ako je f ′(x) > 0 na (a, b).

Funkcija f (x) je opadajuca na intervalu (a, b) ako je f ′(x) < 0 na (a, b).

Stacionarna tacka funkcije f (x) je tacka x = x0 ∈ D ako vazi:

f ′(x0) = 0

Neka je x = x0 stacionarna tacka funkcije f (x). Ako f ′(x) menja znak uokolini tacke x0 onda je x = x0 lokalna ekstremna vrednost funkcije f (x) ito:

tacka x = x0 je lokalni minimum funkcije f (x) ako je f ′′(x0) > 0,

tacka x = x0 je lokalni maksimum funkcije f (x) ako je f ′′(x0) < 0,

ako je f ′′(x0) = 0 potrebna su dalja ispitivanja.






f ′(x0) = 0










f ′(x0) = 0










f ′(x0) = 0










f ′(x0) = 0










f ′(x0) = 0










f ′(x0) = 0










f ′(x0) = 0






Zadatak 88.

Odrediti domen, intervale monotonosti i ekstremne vrednosti funkcije

f (x) = x3 − 6x2 + 9x − 1 .

D =

R f ′(x) = 3x2 − 12x + 9

MONOTONOST: f (x) ↗ za x ∈ (−∞, 1) ∪ (3,+∞)

f (x) ↘ za x ∈ (1, 3)

MINIMUMI: x = 3 MAKSIMUMI: x = 1


Zadatak 88.


f (x) = x3 − 6x2 + 9x − 1 .

D =

R f ′(x) = 3x2 − 12x + 9


f (x) ↘ za x ∈ (1, 3)



Zadatak 88.


f (x) = x3 − 6x2 + 9x − 1 .

D =

R f ′(x) = 3x2 − 12x + 9


f (x) ↘ za x ∈ (1, 3)



Zadatak 88.


f (x) = x3 − 6x2 + 9x − 1 .

D =

R f ′(x) = 3x2 − 12x + 9


f (x) ↘ za x ∈ (1, 3)



Zadatak 88.


f (x) = x3 − 6x2 + 9x − 1 .

D = R f ′(x) =

3x2 − 12x + 9


f (x) ↘ za x ∈ (1, 3)



Zadatak 88.


f (x) = x3 − 6x2 + 9x − 1 .

D = R f ′(x) = 3x2 − 12x + 9

MONOTONOST: f (x) ↗ za x ∈

(−∞, 1) ∪ (3,+∞)

f (x) ↘ za x ∈ (1, 3)



Zadatak 88.


f (x) = x3 − 6x2 + 9x − 1 .

D = R f ′(x) = 3x2 − 12x + 9


f (x) ↘ za x ∈

(1, 3)



Zadatak 88.


f (x) = x3 − 6x2 + 9x − 1 .

D = R f ′(x) = 3x2 − 12x + 9


f (x) ↘ za x ∈ (1, 3)

MINIMUMI:

x = 3 MAKSIMUMI: x = 1


Zadatak 88.


f (x) = x3 − 6x2 + 9x − 1 .

D = R f ′(x) = 3x2 − 12x + 9


f (x) ↘ za x ∈ (1, 3)

MINIMUMI: x = 3 MAKSIMUMI:

x = 1


Zadatak 88.


f (x) = x3 − 6x2 + 9x − 1 .

D = R f ′(x) = 3x2 − 12x + 9


f (x) ↘ za x ∈ (1, 3)



Zadatak 90?.


f (x) =2x + 4

x − 4.

D =

R \ {4} f ′(x) = − 12

(x − 4)2

MONOTONOST: f (x) ↗ za x ∈ ∅f (x) ↘ za x ∈ (−∞, 4) ∪ (4,+∞)

MINIMUMI: nema MAKSIMUMI: nema


Zadatak 90?.


f (x) =2x + 4

x − 4.

D =

R \ {4} f ′(x) = − 12

(x − 4)2




Zadatak 90?.


f (x) =2x + 4

x − 4.

D =

R \ {4} f ′(x) = − 12

(x − 4)2




Zadatak 90?.


f (x) =2x + 4

x − 4.

D =

R \ {4} f ′(x) = − 12

(x − 4)2




Zadatak 90?.


f (x) =2x + 4

x − 4.

D = R \ {4} f ′(x) =

− 12

(x − 4)2




Zadatak 90?.


f (x) =2x + 4

x − 4.

D = R \ {4} f ′(x) = − 12

(x − 4)2


∅f (x) ↘ za x ∈ (−∞, 4) ∪ (4,+∞)



Zadatak 90?.


f (x) =2x + 4

x − 4.

D = R \ {4} f ′(x) = − 12

(x − 4)2

MONOTONOST: f (x) ↗ za x ∈ ∅f (x) ↘ za x ∈

(−∞, 4) ∪ (4,+∞)



Zadatak 90?.


f (x) =2x + 4

x − 4.

D = R \ {4} f ′(x) = − 12

(x − 4)2


MINIMUMI:

nema MAKSIMUMI: nema


Zadatak 90?.


f (x) =2x + 4

x − 4.

D = R \ {4} f ′(x) = − 12

(x − 4)2


MINIMUMI: nema MAKSIMUMI:

nema


Zadatak 90?.


f (x) =2x + 4

x − 4.

D = R \ {4} f ′(x) = − 12

(x − 4)2




Zadatak 91?.


f (x) =x2 − 4

x2 + 1.

D =

R f ′(x) =10x

(x2 + 1)2

MONOTONOST: f (x) ↗ za x ∈ (0,+∞)

f (x) ↘ za x ∈ (−∞, 0)

MINIMUMI: x = 0 MAKSIMUMI: nema


Zadatak 91?.


f (x) =x2 − 4

x2 + 1.

D =

R f ′(x) =10x

(x2 + 1)2


f (x) ↘ za x ∈ (−∞, 0)



Zadatak 91?.


f (x) =x2 − 4

x2 + 1.

D =

R f ′(x) =10x

(x2 + 1)2


f (x) ↘ za x ∈ (−∞, 0)



Zadatak 91?.


f (x) =x2 − 4

x2 + 1.

D =

R f ′(x) =10x

(x2 + 1)2


f (x) ↘ za x ∈ (−∞, 0)



Zadatak 91?.


f (x) =x2 − 4

x2 + 1.

D = R f ′(x) =

10x

(x2 + 1)2


f (x) ↘ za x ∈ (−∞, 0)



Zadatak 91?.


f (x) =x2 − 4

x2 + 1.

D = R f ′(x) =10x

(x2 + 1)2


(0,+∞)

f (x) ↘ za x ∈ (−∞, 0)



Zadatak 91?.


f (x) =x2 − 4

x2 + 1.

D = R f ′(x) =10x

(x2 + 1)2


f (x) ↘ za x ∈

(−∞, 0)



Zadatak 91?.


f (x) =x2 − 4

x2 + 1.

D = R f ′(x) =10x

(x2 + 1)2


f (x) ↘ za x ∈ (−∞, 0)

MINIMUMI:

x = 0 MAKSIMUMI: nema


Zadatak 91?.


f (x) =x2 − 4

x2 + 1.

D = R f ′(x) =10x

(x2 + 1)2


f (x) ↘ za x ∈ (−∞, 0)


nema


Zadatak 91?.


f (x) =x2 − 4

x2 + 1.

D = R f ′(x) =10x

(x2 + 1)2


f (x) ↘ za x ∈ (−∞, 0)



Zadatak 92?.


f (x) =3x + 1

(x − 1)2.

D =

R \ {1} f ′(x) =−3x − 5

(x − 1)3

MONOTONOST: f (x) ↗ za x ∈ (−5

3, 1)

f (x) ↘ za x ∈ (−∞,−5

3) ∪ (1,+∞)

MINIMUMI: x = −5

3MAKSIMUMI: nema


Zadatak 92?.


f (x) =3x + 1

(x − 1)2.

D =

R \ {1} f ′(x) =−3x − 5

(x − 1)3


3, 1)

f (x) ↘ za x ∈ (−∞,−5

3) ∪ (1,+∞)

MINIMUMI: x = −5

3MAKSIMUMI: nema


Zadatak 92?.


f (x) =3x + 1

(x − 1)2.

D =

R \ {1} f ′(x) =−3x − 5

(x − 1)3


3, 1)

f (x) ↘ za x ∈ (−∞,−5

3) ∪ (1,+∞)

MINIMUMI: x = −5

3MAKSIMUMI: nema


Zadatak 92?.


f (x) =3x + 1

(x − 1)2.

D =

R \ {1} f ′(x) =−3x − 5

(x − 1)3


3, 1)

f (x) ↘ za x ∈ (−∞,−5

3) ∪ (1,+∞)

MINIMUMI: x = −5

3MAKSIMUMI: nema


Zadatak 92?.


f (x) =3x + 1

(x − 1)2.

D = R \ {1} f ′(x) =

−3x − 5

(x − 1)3


3, 1)

f (x) ↘ za x ∈ (−∞,−5

3) ∪ (1,+∞)

MINIMUMI: x = −5

3MAKSIMUMI: nema


Zadatak 92?.


f (x) =3x + 1

(x − 1)2.

D = R \ {1} f ′(x) =−3x − 5

(x − 1)3


(−5

3, 1)

f (x) ↘ za x ∈ (−∞,−5

3) ∪ (1,+∞)

MINIMUMI: x = −5

3MAKSIMUMI: nema


Zadatak 92?.


f (x) =3x + 1

(x − 1)2.

D = R \ {1} f ′(x) =−3x − 5

(x − 1)3


3, 1)

f (x) ↘ za x ∈

(−∞,−5

3) ∪ (1,+∞)

MINIMUMI: x = −5

3MAKSIMUMI: nema


Zadatak 92?.


f (x) =3x + 1

(x − 1)2.

D = R \ {1} f ′(x) =−3x − 5

(x − 1)3


3, 1)

f (x) ↘ za x ∈ (−∞,−5

3) ∪ (1,+∞)

MINIMUMI:

x = −5

3MAKSIMUMI: nema


Zadatak 92?.


f (x) =3x + 1

(x − 1)2.

D = R \ {1} f ′(x) =−3x − 5

(x − 1)3


3, 1)

f (x) ↘ za x ∈ (−∞,−5

3) ∪ (1,+∞)

MINIMUMI: x = −5

3MAKSIMUMI:

nema


Zadatak 92?.


f (x) =3x + 1

(x − 1)2.

D = R \ {1} f ′(x) =−3x − 5

(x − 1)3


3, 1)

f (x) ↘ za x ∈ (−∞,−5

3) ∪ (1,+∞)

MINIMUMI: x = −5

3MAKSIMUMI: nema


Zadatak 93?.


f (x) =x3

x2 − 1.

D =

R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2

MONOTONOST: f (x) ↗ za x ∈ (−∞,−√

3) ∪ (√

3,+∞)

f (x) ↘ za x ∈ (−√

3,−1)∪ (−1, 1)∪ (1,√

3)

MINIMUMI: x =√

3 MAKSIMUMI: x = −√

3


Zadatak 93?.


f (x) =x3

x2 − 1.

D =

R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2


3) ∪ (√

3,+∞)

f (x) ↘ za x ∈ (−√

3,−1)∪ (−1, 1)∪ (1,√

3)

MINIMUMI: x =√


3


Zadatak 93?.


f (x) =x3

x2 − 1.

D =

R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2


3) ∪ (√

3,+∞)

f (x) ↘ za x ∈ (−√

3,−1)∪ (−1, 1)∪ (1,√

3)

MINIMUMI: x =√


3


Zadatak 93?.


f (x) =x3

x2 − 1.

D =

R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2


3) ∪ (√

3,+∞)

f (x) ↘ za x ∈ (−√

3,−1)∪ (−1, 1)∪ (1,√

3)

MINIMUMI: x =√


3


Zadatak 93?.


f (x) =x3

x2 − 1.

D = R \ {−1, 1} f ′(x) =

x2(x2 − 3)

(x2 − 1)2


3) ∪ (√

3,+∞)

f (x) ↘ za x ∈ (−√

3,−1)∪ (−1, 1)∪ (1,√

3)

MINIMUMI: x =√


3


Zadatak 93?.


f (x) =x3

x2 − 1.

D = R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2


(−∞,−√

3) ∪ (√

3,+∞)

f (x) ↘ za x ∈ (−√

3,−1)∪ (−1, 1)∪ (1,√

3)

MINIMUMI: x =√


3


Zadatak 93?.


f (x) =x3

x2 − 1.

D = R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2


3) ∪ (√

3,+∞)

f (x) ↘ za x ∈

(−√

3,−1)∪ (−1, 1)∪ (1,√

3)

MINIMUMI: x =√


3


Zadatak 93?.


f (x) =x3

x2 − 1.

D = R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2


3) ∪ (√

3,+∞)

f (x) ↘ za x ∈ (−√

3,−1)∪ (−1, 1)∪ (1,√

3)

MINIMUMI:

x =√


3


Zadatak 93?.


f (x) =x3

x2 − 1.

D = R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2


3) ∪ (√

3,+∞)

f (x) ↘ za x ∈ (−√

3,−1)∪ (−1, 1)∪ (1,√

3)

MINIMUMI: x =√

3 MAKSIMUMI:

x = −√

3


Zadatak 93?.


f (x) =x3

x2 − 1.

D = R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2


3) ∪ (√

3,+∞)

f (x) ↘ za x ∈ (−√

3,−1)∪ (−1, 1)∪ (1,√

3)

MINIMUMI: x =√


3


Zadatak 102.


f (x) = x2ex .

D =

R f ′(x) = x(x + 2)ex

MONOTONOST: f (x) ↗ za x ∈ (−∞,−2) ∪ (0,+∞)

f (x) ↘ za x ∈ (−2, 0)

MINIMUMI: x = 0 MAKSIMUMI: x = −2


Zadatak 102.


f (x) = x2ex .

D =

R f ′(x) = x(x + 2)ex


f (x) ↘ za x ∈ (−2, 0)



Zadatak 102.


f (x) = x2ex .

D =

R f ′(x) = x(x + 2)ex


f (x) ↘ za x ∈ (−2, 0)



Zadatak 102.


f (x) = x2ex .

D =

R f ′(x) = x(x + 2)ex


f (x) ↘ za x ∈ (−2, 0)



Zadatak 102.


f (x) = x2ex .

D = R f ′(x) =

x(x + 2)ex


f (x) ↘ za x ∈ (−2, 0)



Zadatak 102.


f (x) = x2ex .

D = R f ′(x) = x(x + 2)ex


(−∞,−2) ∪ (0,+∞)

f (x) ↘ za x ∈ (−2, 0)



Zadatak 102.


f (x) = x2ex .

D = R f ′(x) = x(x + 2)ex


f (x) ↘ za x ∈

(−2, 0)



Zadatak 102.


f (x) = x2ex .

D = R f ′(x) = x(x + 2)ex


f (x) ↘ za x ∈ (−2, 0)

MINIMUMI:

x = 0 MAKSIMUMI: x = −2


Zadatak 102.


f (x) = x2ex .

D = R f ′(x) = x(x + 2)ex


f (x) ↘ za x ∈ (−2, 0)


x = −2


Zadatak 102.


f (x) = x2ex .

D = R f ′(x) = x(x + 2)ex


f (x) ↘ za x ∈ (−2, 0)



Zadatak 114.


f (x) = ln

(x − 1

x − 2

).

D =

(−∞, 1) ∪ (2,+∞) f ′(x) =−1

(x − 1)(x − 2)




Zadatak 114.


f (x) = ln

(x − 1

x − 2

).

D =

(−∞, 1) ∪ (2,+∞) f ′(x) =−1

(x − 1)(x − 2)




Zadatak 114.


f (x) = ln

(x − 1

x − 2

).

D =

(−∞, 1) ∪ (2,+∞) f ′(x) =−1

(x − 1)(x − 2)




Zadatak 114.


f (x) = ln

(x − 1

x − 2

).

D =

(−∞, 1) ∪ (2,+∞) f ′(x) =−1

(x − 1)(x − 2)




Zadatak 114.


f (x) = ln

(x − 1

x − 2

).

D = (−∞, 1) ∪ (2,+∞) f ′(x) =

−1

(x − 1)(x − 2)




Zadatak 114.


f (x) = ln

(x − 1

x − 2

).

D = (−∞, 1) ∪ (2,+∞) f ′(x) =−1

(x − 1)(x − 2)


∅f (x) ↘ za x ∈ (−∞, 1) ∪ (2,+∞)



Zadatak 114.


f (x) = ln

(x − 1

x − 2

).

D = (−∞, 1) ∪ (2,+∞) f ′(x) =−1

(x − 1)(x − 2)

MONOTONOST: f (x) ↗ za x ∈ ∅f (x) ↘ za x ∈

(−∞, 1) ∪ (2,+∞)



Zadatak 114.


f (x) = ln

(x − 1

x − 2

).

D = (−∞, 1) ∪ (2,+∞) f ′(x) =−1

(x − 1)(x − 2)


MINIMUMI:

nema MAKSIMUMI: nema


Zadatak 114.


f (x) = ln

(x − 1

x − 2

).

D = (−∞, 1) ∪ (2,+∞) f ′(x) =−1

(x − 1)(x − 2)


MINIMUMI: nema MAKSIMUMI:

nema


Zadatak 114.


f (x) = ln

(x − 1

x − 2

).

D = (−∞, 1) ∪ (2,+∞) f ′(x) =−1

(x − 1)(x − 2)




Zadaci

Zadatak 89. Odrediti domen, intervale monotonosti i ekstremne vrednostifunkcije f (x) = (x + 1)(x + 2)(x + 3)− 2x .

Zadatak 90. Odrediti domen, intervale monotonosti i ekstremne vrednosti

funkcije f (x) =x − 1

x + 1.


funkcije f (x) =2x

2x2 + 5x + 2.

Zadatak 110. Odrediti domen, intervale monotonosti i ekstremne

vrednosti funkcije f (x) =ex

x.

Zadatak 115. Odrediti domen, intervale monotonosti i ekstremnevrednosti funkcije f (x) = x ln x .


Zadaci




x + 1.


funkcije f (x) =2x

2x2 + 5x + 2.



x.



Zadaci




x + 1.


funkcije f (x) =2x

2x2 + 5x + 2.



x.



Zadaci




x + 1.


funkcije f (x) =2x

2x2 + 5x + 2.



x.



Zadaci




x + 1.


funkcije f (x) =2x

2x2 + 5x + 2.



x.



Zadaci




x + 1.


funkcije f (x) =2x

2x2 + 5x + 2.



x.



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Ekstremne vrednosti funkcije - formule i zadaci -