Applications of derivatives

1103266407

Level:

Our task is to support a square tarp sized

4 m \times 4 m

in the centers of its opposite sides to create a hay roof, as shown in the picture. What should be the height

v

of the pillars to make the shelter space (volume) as large as possible?

\sqrt{2} m

\frac{\sqrt{2}}{2} m

2 m

\sqrt{3} m

\frac{\sqrt{3}}{2} m

2010010901

Level:

Find the minimum value of the function

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

on the interval

[0; 2]

- 3

- 1

- \frac{3}{4}

- 7

2010010902

Level:

Find the minimum value of the function

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

on the interval

[- 2; 0]

0

2

\frac{9}{4}

- 4

2010010903

Level:

Find the minimum value of the function

f (x) = 2 x - 4

on the interval

(0; 4)

The function

f

does not have the minimum on the interval

(0; 4)

- 4

4

- 6

2010010904

Level:

Find the minimum value of the function

f (x) = - 3 x + 6

on the interval

(0; 4)

The function

f

does not have the minimum on the interval

(0; 4)

6

- 6

- 9

2010010905

Level:

Find the maximum of the function

g (x) = 3 x - \frac{1}{1 - 3 x}

when

x \in [- 1; \frac{1}{9}]

x = 0

x = - \frac{13}{4}

x = - \frac{7}{6}

The function

g

does not have the maximum on the interval

[- 1; \frac{1}{9}]

2010010906

Level:

Find the maximum of the function

g (x) = \frac{- 1}{1 - 2 x} + 2 x

when

x \in [- 2; \frac{1}{4}]

x = 0

x = - \frac{17}{4}

x = - \frac{3}{2}

The function

g

does not have the maximum on the interval

[- 2; \frac{1}{4}]

2010010907

Level:

How many of these functions do have a global maximum on the interval

(- \infty; - 1]

at the point

x = - 1

f (x) = x + \frac{1}{x} + 2

g (x) = - x^{2} + 6 x - 9

h (x) = 2 x - 3

3

1

2

0

2010010908

Level:

How many of these functions do have a global minimum on the interval

[- 1; \infty)

at the point

x = - 1

f (x) = x + \frac{1}{x + 2}

g (x) = x^{2} + 4 x + 4

h (x) = 3 x + 1

3

1

2

0

2010010909

Level:

Find the maximum of the function

g (x) = \frac{x^{4}}{2} - 4 x^{2} + 2

when

x \in [- 2; 3]

x = 3

x = - 2

x = 0

The function

g

does not have the maximum on the interval

[- 2; 3]

Applications of derivatives

1103266407

2010010901

2010010902

2010010903

2010010904

2010010905

2010010906

2010010907

2010010908

2010010909

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