prevojne taˇcke funkcije - formule i...

Prevojne tacke funkcije- formule i zadaci -

2010/2011

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

Konveksnost, konkavnost i prevojne tacke funkcije

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

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

Neka u tacki x = x0 vazi f ′′(x0) = 0. Ako f ′′(x) menja znak u okolinitacke x0 onda je x = x0 prevojna tacka funkcije f (x).

Zadatak 106.

Odrediti domen, intervale konkavnosti, konveksnosti i prevojne tackefunkcije

f (x) = 3x5 − 10x3 + 7 .

R f ′(x) = 15x2(x2 − 2) f ′′(x) = 60x(x − 1)(x + 1)

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

PREVOJNE TACKE: x = −1, x = 0, x = 1

Zadatak 106.

f (x) = 3x5 − 10x3 + 7 .

R f ′(x) = 15x2(x2 − 2) f ′′(x) = 60x(x − 1)(x + 1)

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 106.

f (x) = 3x5 − 10x3 + 7 .

R f ′(x) = 15x2(x2 − 2) f ′′(x) = 60x(x − 1)(x + 1)

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 106.

f (x) = 3x5 − 10x3 + 7 .

R f ′(x) = 15x2(x2 − 2) f ′′(x) = 60x(x − 1)(x + 1)

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 106.

f (x) = 3x5 − 10x3 + 7 .

D = R f ′(x) =

15x2(x2 − 2) f ′′(x) = 60x(x − 1)(x + 1)

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 106.

f (x) = 3x5 − 10x3 + 7 .

D = R f ′(x) = 15x2(x2 − 2) f ′′(x) =

60x(x − 1)(x + 1)

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 106.

f (x) = 3x5 − 10x3 + 7 .

D = R f ′(x) = 15x2(x2 − 2) f ′′(x) = 60x(x − 1)(x + 1)

f (x) ^ za x ∈

(−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 106.

f (x) = 3x5 − 10x3 + 7 .

D = R f ′(x) = 15x2(x2 − 2) f ′′(x) = 60x(x − 1)(x + 1)

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈

(−∞,−1) ∪ (0, 1)

Zadatak 106.

f (x) = 3x5 − 10x3 + 7 .

D = R f ′(x) = 15x2(x2 − 2) f ′′(x) = 60x(x − 1)(x + 1)

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

PREVOJNE TACKE:

x = −1, x = 0, x = 1

Zadatak 106.

f (x) = 3x5 − 10x3 + 7 .

D = R f ′(x) = 15x2(x2 − 2) f ′′(x) = 60x(x − 1)(x + 1)

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 108?.

f (x) =2x + 4

x − 4.

R \ {4} f ′(x) = − 12

(x − 4)2f ′′(x) =

(x − 4)3

f (x) ^ za x ∈ (4,+∞) f (x) _ za x ∈ (−∞, 4)

PREVOJNE TACKE: nema

Zadatak 108?.

f (x) =2x + 4

x − 4.

R \ {4} f ′(x) = − 12

(x − 4)2f ′′(x) =

(x − 4)3

f (x) ^ za x ∈ (4,+∞) f (x) _ za x ∈ (−∞, 4)

Zadatak 108?.

f (x) =2x + 4

x − 4.

R \ {4} f ′(x) = − 12

(x − 4)2f ′′(x) =

(x − 4)3

f (x) ^ za x ∈ (4,+∞) f (x) _ za x ∈ (−∞, 4)

Zadatak 108?.

f (x) =2x + 4

x − 4.

R \ {4} f ′(x) = − 12

(x − 4)2f ′′(x) =

(x − 4)3

f (x) ^ za x ∈ (4,+∞) f (x) _ za x ∈ (−∞, 4)

Zadatak 108?.

f (x) =2x + 4

x − 4.

D = R \ {4} f ′(x) =

− 12

(x − 4)2f ′′(x) =

(x − 4)3

f (x) ^ za x ∈ (4,+∞) f (x) _ za x ∈ (−∞, 4)

Zadatak 108?.

f (x) =2x + 4

x − 4.

D = R \ {4} f ′(x) = − 12

(x − 4)2f ′′(x) =

(x − 4)3

f (x) ^ za x ∈ (4,+∞) f (x) _ za x ∈ (−∞, 4)

Zadatak 108?.

f (x) =2x + 4

x − 4.

D = R \ {4} f ′(x) = − 12

(x − 4)2f ′′(x) =

(x − 4)3

f (x) ^ za x ∈

(4,+∞) f (x) _ za x ∈ (−∞, 4)

Zadatak 108?.

f (x) =2x + 4

x − 4.

D = R \ {4} f ′(x) = − 12

(x − 4)2f ′′(x) =

(x − 4)3

f (x) ^ za x ∈ (4,+∞) f (x) _ za x ∈

(−∞, 4)

Zadatak 108?.

f (x) =2x + 4

x − 4.

D = R \ {4} f ′(x) = − 12

(x − 4)2f ′′(x) =

(x − 4)3

f (x) ^ za x ∈ (4,+∞) f (x) _ za x ∈ (−∞, 4)

PREVOJNE TACKE:

Zadatak 108?.

f (x) =2x + 4

x − 4.

D = R \ {4} f ′(x) = − 12

(x − 4)2f ′′(x) =

(x − 4)3

f (x) ^ za x ∈ (4,+∞) f (x) _ za x ∈ (−∞, 4)

Zadatak 109?.

f (x) =x2 − 4

x2 + 1.

R f ′(x) =10x

(x2 + 1)2f ′′(x) =

10− 30x2

(x2 + 1)3

f (x) ^ za x ∈ (− 1√3, 1√

3) f (x) _ za x ∈ (−∞,− 1√

3) ∪ ( 1√

3,+∞)

PREVOJNE TACKE: x = − 1√3, x =

Zadatak 109?.

f (x) =x2 − 4

x2 + 1.

R f ′(x) =10x

(x2 + 1)2f ′′(x) =

10− 30x2

(x2 + 1)3

f (x) ^ za x ∈ (− 1√3, 1√

3) f (x) _ za x ∈ (−∞,− 1√

3) ∪ ( 1√

3,+∞)

Zadatak 109?.

f (x) =x2 − 4

x2 + 1.

R f ′(x) =10x

(x2 + 1)2f ′′(x) =

10− 30x2

(x2 + 1)3

f (x) ^ za x ∈ (− 1√3, 1√

3) f (x) _ za x ∈ (−∞,− 1√

3) ∪ ( 1√

3,+∞)

Zadatak 109?.

f (x) =x2 − 4

x2 + 1.

R f ′(x) =10x

(x2 + 1)2f ′′(x) =

10− 30x2

(x2 + 1)3

f (x) ^ za x ∈ (− 1√3, 1√

3) f (x) _ za x ∈ (−∞,− 1√

3) ∪ ( 1√

3,+∞)

Zadatak 109?.

f (x) =x2 − 4

x2 + 1.

D = R f ′(x) =

(x2 + 1)2f ′′(x) =

10− 30x2

(x2 + 1)3

f (x) ^ za x ∈ (− 1√3, 1√

3) f (x) _ za x ∈ (−∞,− 1√

3) ∪ ( 1√

3,+∞)

Zadatak 109?.

f (x) =x2 − 4

x2 + 1.

D = R f ′(x) =10x

(x2 + 1)2f ′′(x) =

10− 30x2

(x2 + 1)3

f (x) ^ za x ∈ (− 1√3, 1√

3) f (x) _ za x ∈ (−∞,− 1√

3) ∪ ( 1√

3,+∞)

Zadatak 109?.

f (x) =x2 − 4

x2 + 1.

D = R f ′(x) =10x

(x2 + 1)2f ′′(x) =

10− 30x2

(x2 + 1)3

f (x) ^ za x ∈

(− 1√3, 1√

3) f (x) _ za x ∈ (−∞,− 1√

3) ∪ ( 1√

3,+∞)

Zadatak 109?.

f (x) =x2 − 4

x2 + 1.

D = R f ′(x) =10x

(x2 + 1)2f ′′(x) =

10− 30x2

(x2 + 1)3

f (x) ^ za x ∈ (− 1√3, 1√

3) f (x) _ za x ∈

(−∞,− 1√3) ∪ ( 1√

3,+∞)

Zadatak 109?.

f (x) =x2 − 4

x2 + 1.

D = R f ′(x) =10x

(x2 + 1)2f ′′(x) =

10− 30x2

(x2 + 1)3

f (x) ^ za x ∈ (− 1√3, 1√

3) f (x) _ za x ∈ (−∞,− 1√

3) ∪ ( 1√

3,+∞)

PREVOJNE TACKE:

x = − 1√3, x =

Zadatak 109?.

f (x) =x2 − 4

x2 + 1.

D = R f ′(x) =10x

(x2 + 1)2f ′′(x) =

10− 30x2

(x2 + 1)3

f (x) ^ za x ∈ (− 1√3, 1√

3) f (x) _ za x ∈ (−∞,− 1√

3) ∪ ( 1√

3,+∞)

Zadatak 107?.

f (x) =3x + 1

(x − 1)2.

R \ {1} f ′(x) =−3x − 5

(x − 1)3f ′′(x) =

6(x + 3)

(x − 1)4

f (x) ^ za x ∈ (−3, 1) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−3)

PREVOJNE TACKE: x = −3

Zadatak 107?.

f (x) =3x + 1

(x − 1)2.

R \ {1} f ′(x) =−3x − 5

(x − 1)3f ′′(x) =

6(x + 3)

(x − 1)4

f (x) ^ za x ∈ (−3, 1) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−3)

Zadatak 107?.

f (x) =3x + 1

(x − 1)2.

R \ {1} f ′(x) =−3x − 5

(x − 1)3f ′′(x) =

6(x + 3)

(x − 1)4

f (x) ^ za x ∈ (−3, 1) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−3)

Zadatak 107?.

f (x) =3x + 1

(x − 1)2.

R \ {1} f ′(x) =−3x − 5

(x − 1)3f ′′(x) =

6(x + 3)

(x − 1)4

f (x) ^ za x ∈ (−3, 1) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−3)

Zadatak 107?.

f (x) =3x + 1

(x − 1)2.

D = R \ {1} f ′(x) =

−3x − 5

(x − 1)3f ′′(x) =

6(x + 3)

(x − 1)4

f (x) ^ za x ∈ (−3, 1) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−3)

Zadatak 107?.

f (x) =3x + 1

(x − 1)2.

D = R \ {1} f ′(x) =−3x − 5

(x − 1)3f ′′(x) =

6(x + 3)

(x − 1)4

f (x) ^ za x ∈ (−3, 1) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−3)

Zadatak 107?.

f (x) =3x + 1

(x − 1)2.

D = R \ {1} f ′(x) =−3x − 5

(x − 1)3f ′′(x) =

6(x + 3)

(x − 1)4

f (x) ^ za x ∈

(−3, 1) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−3)

Zadatak 107?.

f (x) =3x + 1

(x − 1)2.

D = R \ {1} f ′(x) =−3x − 5

(x − 1)3f ′′(x) =

6(x + 3)

(x − 1)4

f (x) ^ za x ∈ (−3, 1) ∪ (1,+∞) f (x) _ za x ∈

(−∞,−3)

Zadatak 107?.

f (x) =3x + 1

(x − 1)2.

D = R \ {1} f ′(x) =−3x − 5

(x − 1)3f ′′(x) =

6(x + 3)

(x − 1)4

f (x) ^ za x ∈ (−3, 1) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−3)

PREVOJNE TACKE:

x = −3

Zadatak 107?.

f (x) =3x + 1

(x − 1)2.

D = R \ {1} f ′(x) =−3x − 5

(x − 1)3f ′′(x) =

6(x + 3)

(x − 1)4

f (x) ^ za x ∈ (−3, 1) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−3)

Zadatak 109?.

f (x) =x3

x2 − 1.

R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2f ′′(x) =

2x(x2 + 3)

(x2 − 1)3

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

PREVOJNE TACKE: x = 0

Zadatak 109?.

f (x) =x3

x2 − 1.

R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2f ′′(x) =

2x(x2 + 3)

(x2 − 1)3

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 109?.

f (x) =x3

x2 − 1.

R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2f ′′(x) =

2x(x2 + 3)

(x2 − 1)3

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 109?.

f (x) =x3

x2 − 1.

R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2f ′′(x) =

2x(x2 + 3)

(x2 − 1)3

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 109?.

f (x) =x3

x2 − 1.

D = R \ {−1, 1} f ′(x) =

x2(x2 − 3)

(x2 − 1)2f ′′(x) =

2x(x2 + 3)

(x2 − 1)3

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 109?.

f (x) =x3

x2 − 1.

D = R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2f ′′(x) =

2x(x2 + 3)

(x2 − 1)3

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 109?.

f (x) =x3

x2 − 1.

D = R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2f ′′(x) =

2x(x2 + 3)

(x2 − 1)3

f (x) ^ za x ∈

(−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 109?.

f (x) =x3

x2 − 1.

D = R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2f ′′(x) =

2x(x2 + 3)

(x2 − 1)3

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈

(−∞,−1) ∪ (0, 1)

Zadatak 109?.

f (x) =x3

x2 − 1.

D = R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2f ′′(x) =

2x(x2 + 3)

(x2 − 1)3

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

PREVOJNE TACKE:

Zadatak 109?.

f (x) =x3

x2 − 1.

D = R \ {−1, 1} f ′(x) =x2(x2 − 3)

(x2 − 1)2f ′′(x) =

2x(x2 + 3)

(x2 − 1)3

f (x) ^ za x ∈ (−1, 0) ∪ (1,+∞) f (x) _ za x ∈ (−∞,−1) ∪ (0, 1)

Zadatak 102?.

f (x) = x2ex .

R f ′(x) = x(x + 2)ex f ′′(x) = (x2 + 4x + 2)ex

f (x) ^ za x ∈ (−∞,−2−√

2) ∪ (−2 +√

2,+∞)f (x) _ za x ∈ (−2−

√2,−2 +

PREVOJNE TACKE: x = −2−√

2, x = −2 +√

Zadatak 102?.

f (x) = x2ex .

R f ′(x) = x(x + 2)ex f ′′(x) = (x2 + 4x + 2)ex

f (x) ^ za x ∈ (−∞,−2−√

2) ∪ (−2 +√

2,+∞)f (x) _ za x ∈ (−2−

√2,−2 +

2, x = −2 +√

Zadatak 102?.

f (x) = x2ex .

R f ′(x) = x(x + 2)ex f ′′(x) = (x2 + 4x + 2)ex

f (x) ^ za x ∈ (−∞,−2−√

2) ∪ (−2 +√

2,+∞)f (x) _ za x ∈ (−2−

√2,−2 +

2, x = −2 +√

Zadatak 102?.

f (x) = x2ex .

R f ′(x) = x(x + 2)ex f ′′(x) = (x2 + 4x + 2)ex

f (x) ^ za x ∈ (−∞,−2−√

2) ∪ (−2 +√

2,+∞)f (x) _ za x ∈ (−2−

√2,−2 +

2, x = −2 +√

Zadatak 102?.

f (x) = x2ex .

D = R f ′(x) =

x(x + 2)ex f ′′(x) = (x2 + 4x + 2)ex

f (x) ^ za x ∈ (−∞,−2−√

2) ∪ (−2 +√

2,+∞)f (x) _ za x ∈ (−2−

√2,−2 +

2, x = −2 +√

Zadatak 102?.

f (x) = x2ex .

D = R f ′(x) = x(x + 2)ex f ′′(x) =

(x2 + 4x + 2)ex

f (x) ^ za x ∈ (−∞,−2−√

2) ∪ (−2 +√

2,+∞)f (x) _ za x ∈ (−2−

√2,−2 +

2, x = −2 +√

Zadatak 102?.

f (x) = x2ex .

D = R f ′(x) = x(x + 2)ex f ′′(x) = (x2 + 4x + 2)ex

f (x) ^ za x ∈

(−∞,−2−√

2) ∪ (−2 +√

2,+∞)f (x) _ za x ∈ (−2−

√2,−2 +

2, x = −2 +√

Zadatak 102?.

f (x) = x2ex .

D = R f ′(x) = x(x + 2)ex f ′′(x) = (x2 + 4x + 2)ex

f (x) ^ za x ∈ (−∞,−2−√

2) ∪ (−2 +√

2,+∞)f (x) _ za x ∈

(−2−√

2,−2 +√

2, x = −2 +√

Zadatak 102?.

f (x) = x2ex .

D = R f ′(x) = x(x + 2)ex f ′′(x) = (x2 + 4x + 2)ex

f (x) ^ za x ∈ (−∞,−2−√

2) ∪ (−2 +√

2,+∞)f (x) _ za x ∈ (−2−

√2,−2 +

PREVOJNE TACKE:

x = −2−√

2, x = −2 +√

Zadatak 102?.

f (x) = x2ex .

D = R f ′(x) = x(x + 2)ex f ′′(x) = (x2 + 4x + 2)ex

f (x) ^ za x ∈ (−∞,−2−√

2) ∪ (−2 +√

2,+∞)f (x) _ za x ∈ (−2−

√2,−2 +

2, x = −2 +√

Zadatak 114.

f (x) = ln

(x − 1

x − 2

(−∞, 1) ∪ (2,+∞) f ′(x) = −1(x−1)(x−2) f ′′(x) = 2x−3

((x−1)(x−2))2

f (x) ^ za x ∈ (2,+∞) f (x) _ za x ∈ (−∞, 1)

Zadatak 114.

f (x) = ln

(x − 1

x − 2

(−∞, 1) ∪ (2,+∞) f ′(x) = −1(x−1)(x−2) f ′′(x) = 2x−3

((x−1)(x−2))2

f (x) ^ za x ∈ (2,+∞) f (x) _ za x ∈ (−∞, 1)

Zadatak 114.

f (x) = ln

(x − 1

x − 2

(−∞, 1) ∪ (2,+∞) f ′(x) = −1(x−1)(x−2) f ′′(x) = 2x−3

((x−1)(x−2))2

f (x) ^ za x ∈ (2,+∞) f (x) _ za x ∈ (−∞, 1)

Zadatak 114.

f (x) = ln

(x − 1

x − 2

(−∞, 1) ∪ (2,+∞) f ′(x) = −1(x−1)(x−2) f ′′(x) = 2x−3

((x−1)(x−2))2

f (x) ^ za x ∈ (2,+∞) f (x) _ za x ∈ (−∞, 1)

Zadatak 114.

f (x) = ln

(x − 1

x − 2

D = (−∞, 1) ∪ (2,+∞) f ′(x) =

−1(x−1)(x−2) f ′′(x) = 2x−3

((x−1)(x−2))2

f (x) ^ za x ∈ (2,+∞) f (x) _ za x ∈ (−∞, 1)

Zadatak 114.

f (x) = ln

(x − 1

x − 2

D = (−∞, 1) ∪ (2,+∞) f ′(x) = −1(x−1)(x−2) f ′′(x) =

2x−3((x−1)(x−2))2

f (x) ^ za x ∈ (2,+∞) f (x) _ za x ∈ (−∞, 1)

Zadatak 114.

f (x) = ln

(x − 1

x − 2

D = (−∞, 1) ∪ (2,+∞) f ′(x) = −1(x−1)(x−2) f ′′(x) = 2x−3

((x−1)(x−2))2

f (x) ^ za x ∈

(2,+∞) f (x) _ za x ∈ (−∞, 1)

Zadatak 114.

f (x) = ln

(x − 1

x − 2

D = (−∞, 1) ∪ (2,+∞) f ′(x) = −1(x−1)(x−2) f ′′(x) = 2x−3

((x−1)(x−2))2

f (x) ^ za x ∈ (2,+∞) f (x) _ za x ∈

(−∞, 1)

Zadatak 114.

f (x) = ln

(x − 1

x − 2

D = (−∞, 1) ∪ (2,+∞) f ′(x) = −1(x−1)(x−2) f ′′(x) = 2x−3

((x−1)(x−2))2

f (x) ^ za x ∈ (2,+∞) f (x) _ za x ∈ (−∞, 1)

PREVOJNE TACKE:

Zadatak 114.

f (x) = ln

(x − 1

x − 2

D = (−∞, 1) ∪ (2,+∞) f ′(x) = −1(x−1)(x−2) f ′′(x) = 2x−3

((x−1)(x−2))2

f (x) ^ za x ∈ (2,+∞) f (x) _ za x ∈ (−∞, 1)

Zadaci

Zadatak 89. Odrediti domen, intervale konkavnosti, konveksnosti iprevojne tacke funkcije f (x) = (x + 1)(x + 2)(x + 3)− 2x .

Zadatak 90. Odrediti domen, intervale konkavnosti, konveksnosti i

prevojne tacke funkcije f (x) =x − 1

x + 1.

prevojne tacke funkcije f (x) =2x

2x2 + 5x + 2.

prevojne tacke funkcije f (x) =ex

Zadatak 115. Odrediti domen, intervale konkavnosti, konveksnosti iprevojne tacke funkcije f (x) = x ln x .

Zadaci

x + 1.

2x2 + 5x + 2.

Zadaci

x + 1.

2x2 + 5x + 2.

Zadaci

x + 1.

2x2 + 5x + 2.

Zadaci

x + 1.

2x2 + 5x + 2.

Zadaci

x + 1.

2x2 + 5x + 2.

prevojne taˇcke funkcije - formule i...

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