Analyzing function behavior

1003261905

Level:

Find the local extrema of the function

f (x) = x - \ln (1 + x) .

the local minimum at

x = 0

the local minimum at

x = 0

, the local maximum at

x = - 1

the local maximum at

x = 0

the local maximum at

x = 0

, the local minimum at

x = - 1

do not exist

1003261906

Level:

Determine the values of

a

and

b

(

a

b \in R

) such that the function

f (x) = a x^{3} + b x^{2} + 1

has a local extremum of

9

x = - 2

a = 2

b = 6

a = - 2

b = 6

a = - 2

b = - 6

a = 2

b = - 6

1003261907

Level:

Determine the values of

m

and

n

(

m

n \in R

) such that the function

f (x) = x^{4} + m x^{3} - n + 2

has a local minimum of

- 30

x = 3

m = - 4

n = 5

m = 4

n = 5

m = 4

n = 140

m = - 4

n = - 5

1003261908

Level:

Determine all the values of

t

t \in R

, such that the function

f (x) = t x^{3} + (t + 1) x^{2} - (t - 2) x + 3

has local extrema.

t \in R ∖ {\frac{1}{2}}

t \in R

t \in (- \frac{1}{2}; \frac{1}{2})

t \in (- \infty; - \frac{1}{2}) \cup (\frac{1}{2}; \infty)

1003261909

Level:

Determine all the values of

a

a \in R

, such that the function

f (x) = \frac{a^{2} - 1}{3} x^{3} + (a - 1) x^{2} + 2 x + 1

does not have any local extrema.

a \in (- \infty; - 3] \cup [1; \infty)

a \in (- \infty; - 3) \cup (1; \infty)

a \in (- 3; 1)

a \in [- 3; 1]

1103163602

Level:

The graph of

f^{'}

is given in the figure. Find the interval where

f

is an increasing function. (The function

f^{'}

is the derivative of the function

f

(- 1; 1)

(- 3; - 1)

(2; 4)

(0; 2)

1103163603

Level:

The graph of

f^{'}

is given in the figure. Find the interval where

f

is an increasing function. (The function

f^{'}

is the derivative of the function

f

(1; 2)

(- 1; 1)

(1; 3)

(- 1; 0)

1103163604

Level:

The graph of

f^{'}

is given in the figure. Find the interval where

f

is a decreasing function. (The function

f^{'}

is the derivative of the function

f

(- 3; - 2)

(- 1; 1)

(0; 2)

(- 1; 2)

1103163605

Level:

The graph of

f^{'}

is given in the figure. Find the interval where

f

is a decreasing function. (The function

f^{'}

is the derivative of the function

f

(2; 4)

(- 1; 1)

(1; 3)

(- 4; - 2)

1103163606

Level:

The graph of

f^{'}

is given in the figure. Find the local extrema of

f

. (The function

f^{'}

is the derivative of the function

f

local minimum at

x = 0

, local maxima at

x_{1} = - 2

and

x_{2} = 3

local minimum at

x = - 1

, local maximum at

x = 2

local minima at

x_{1} = - 2

and

x_{2} = 3

, local maximum at

x = 0

local minima at

x_{1} = - 2

and

x_{2} = 0

, local maximum at

x = 3

local minimum at

x = - 2

, local maxima at

x_{1} = 0

and

x_{2} = 2

Analyzing function behavior

1003261905

1003261906

1003261907

1003261908

1003261909

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1103163604

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