Lesson test: Extrema and optimisation

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Extrema and optimisation

1.[2p]
What is the global maximum of $f (x) = x^{3} - 6 x^{2} + 9 x + 2$ on $[0, 5]$ ?

CorrectNot quite: 22

2.[2p]
What is the global minimum of the same function on the same interval?

CorrectNot quite: 2

3.[3p]
Fermat's theorem says that at an interior local extreme of a differentiable function the derivative is zero. What does it not say?

That a point with zero derivative is an extreme, which $x^{3}$ at the origin disproves

Correct

That extremes exist at all

The answer is: That a point with zero derivative is an extreme, which $x^{3}$ at the origin disproves

That the derivative exists at the extreme

The answer is: That a point with zero derivative is an extreme, which $x^{3}$ at the origin disproves

That the function is continuous

The answer is: That a point with zero derivative is an extreme, which $x^{3}$ at the origin disproves

4.[3p]
An open box is folded from a 30 cm square sheet by cutting a square of side $x$ from each corner. What $x$ , in cm, gives the largest volume?

CorrectNot quite: 5

5.[3p]
What radius, in cm, minimises the metal in a cylindrical can holding 355 mL?

CorrectNot quite: 3.84

6.[2p]
Real drink cans are much taller than the optimum found by that calculation. Why?

The ends are thicker and cost more per unit area, and the can must also suit a hand and a machine, so area is only one term of the real objective

Correct

The calculation is wrong

The answer is: The ends are thicker and cost more per unit area, and the can must also suit a hand and a machine, so area is only one term of the real objective

Manufacturers do not use calculus

The answer is: The ends are thicker and cost more per unit area, and the can must also suit a hand and a machine, so area is only one term of the real objective

The volume of a real can is not 355 mL

The answer is: The ends are thicker and cost more per unit area, and the can must also suit a hand and a machine, so area is only one term of the real objective

7.[2p]
Every point where $f' (c) = 0$ is either a local maximum or a local minimum.

True

The answer is: False

False

Correct

8.[3p]
Light meets water at 30 degrees to the normal, and water has refractive index 1.333. What is the refraction angle, in degrees?

CorrectNot quite: 22.0

9.[3p]
Match each test to what it concludes at a critical point $c$ .

$f'$ changes positive to negative
$f'$ changes negative to positive
$f ′′ (c) > 0$
$f ′′ (c) = 0$

local minimum
local minimum
the test is silent
local maximum

Show the answer

$f'$ changes positive to negative: local maximum $f'$ changes negative to positive: local minimum $f ′′ (c) > 0$ : local minimum $f ′′ (c) = 0$ : the test is silent

1.[2p] What is the global maximum of f(x)=x3−6x2+9x+2 on [0,5]?

2.[2p] What is the global minimum of the same function on the same interval?

3.[3p] Fermat's theorem says that at an interior local extreme of a differentiable function the derivative is zero. What does it not say?

4.[3p] An open box is folded from a 30 cm square sheet by cutting a square of side x from each corner. What x, in cm, gives the largest volume?

5.[3p] What radius, in cm, minimises the metal in a cylindrical can holding 355 mL?

6.[2p] Real drink cans are much taller than the optimum found by that calculation. Why?

7.[2p] Every point where f′(c)=0 is either a local maximum or a local minimum.

8.[3p] Light meets water at 30 degrees to the normal, and water has refractive index 1.333. What is the refraction angle, in degrees?

9.[3p] Match each test to what it concludes at a critical point c.

1.[2p]
What is the global maximum of $f (x) = x^{3} - 6 x^{2} + 9 x + 2$ on $[0, 5]$ ?

2.[2p]
What is the global minimum of the same function on the same interval?

3.[3p]
Fermat's theorem says that at an interior local extreme of a differentiable function the derivative is zero. What does it not say?

4.[3p]
An open box is folded from a 30 cm square sheet by cutting a square of side $x$ from each corner. What $x$ , in cm, gives the largest volume?

5.[3p]
What radius, in cm, minimises the metal in a cylindrical can holding 355 mL?

6.[2p]
Real drink cans are much taller than the optimum found by that calculation. Why?

7.[2p]
Every point where $f' (c) = 0$ is either a local maximum or a local minimum.

8.[3p]
Light meets water at 30 degrees to the normal, and water has refractive index 1.333. What is the refraction angle, in degrees?

9.[3p]
Match each test to what it concludes at a critical point $c$ .