Lesson test: Maxima and minima

Maxima and minima

1.[2p]
The function $f (x, y) = x^{3} - 3 x + y^{2} - 4 y$ has a critical point at $(1, 2)$ . What is $D = f_{x x} f_{y y} - f_{x y}^{2}$ there?

CorrectNot quite: 12

2.[2p]
The same $f (x, y) = x^{3} - 3 x + y^{2} - 4 y$ has exactly one local minimum. What is its value?

CorrectNot quite: -6

3.[2p]
At the origin, $f (x, y) = x^{2} + 6 x y + 4 y^{2}$ has gradient $𝟎$ , with $f_{x x} = 2$ and $f_{y y} = 8$ . What kind of critical point is it?

A saddle, since $D = 16 - 36 < 0$

Correct

A local minimum, since $f_{x x}$ and $f_{y y}$ are both positive

The answer is: A saddle, since $D = 16 - 36 < 0$

A local minimum, since $D = 16 > 0$ and $f_{x x} > 0$

The answer is: A saddle, since $D = 16 - 36 < 0$

The test is inconclusive, since $f_{x y} \neq 0$

The answer is: A saddle, since $D = 16 - 36 < 0$

4.[3p]
Use the second order Taylor polynomial of $f (x, y) = e^{x} \sin y$ at the origin to estimate $f (0.1, 0.2)$ . Give the estimate to two decimal places.

CorrectNot quite: 0.22

5.[2p]
The Mauna Loa annual means for 2014 to 2018, rounded, were $399$ , $401$ , $404$ , $407$ and $409$ ppm. With $x$ counting years from 2014, what is the slope of the least squares line, in ppm per year?

CorrectNot quite: 2.6

6.[2p]
For the same data, what is the intercept $c$ of the least squares line $y = m x + c$ ? Give it to one decimal place.

CorrectNot quite: 398.8

7.[3p]
Find the absolute maximum of $f (x, y) = x^{2} - 4 x + y^{2} - 2 y$ on the square $0 \leq x \leq 3$ , $0 \leq y \leq 3$ .

CorrectNot quite: 3

8.[1p]
Match each function to the behaviour of its critical point at the origin.

$x^{4} + y^{4}$
$x^{4} - y^{4}$
$x^{2} - y^{2}$
$- x^{2} - y^{2}$
points
tags

a saddle confirmed by $D = - 4$
a maximum confirmed by $D = 4$
second-derivative-test, degenerate
2
a saddle the test cannot detect, with $D = 0$
a minimum the test cannot detect, with $D = 0$

Show the answer

$x^{4} + y^{4}$ : a minimum the test cannot detect, with $D = 0$ $x^{4} - y^{4}$ : a saddle the test cannot detect, with $D = 0$ $x^{2} - y^{2}$ : a saddle confirmed by $D = - 4$ $- x^{2} - y^{2}$ : a maximum confirmed by $D = 4$

points: 2 tags: second-derivative-test, degenerate

9.[3p]
Which of these statements are true?

Select all that apply

The least squares line passes through the point $(\overline{x}, \overline{y})$

Correct

The tip of the cone $\sqrt{x^{2} + y^{2}}$ is a critical point even though its gradient does not exist there

Correct

An absolute maximum on a closed and bounded region can occur at a boundary point where $\nabla f \neq 𝟎$

Correct

$f (x, y) = x$ attains an absolute maximum on the open disc $x^{2} + y^{2} < 1$

The answer is: The least squares line passes through the point $(\bar{x}, \bar{y})$, The tip of the cone $\sqrt{x^2 + y^2}$ is a critical point even though its gradient does not exist there, An absolute maximum on a closed and bounded region can occur at a boundary point where $\nabla f \ne \mathbf{0}$

If $D = 0$ at a critical point, the point must be a saddle

1.[2p] The function f(x,y)=x3-3x+y2-4y has a critical point at (1,2). What is D=fxxfyy-fxy2 there?

2.[2p] The same f(x,y)=x3-3x+y2-4y has exactly one local minimum. What is its value?

3.[2p] At the origin, f(x,y)=x2+6xy+4y2 has gradient 𝟎, with fxx=2 and fyy=8. What kind of critical point is it?

4.[3p] Use the second order Taylor polynomial of f(x,y)=exsiny at the origin to estimate f(0.1,0.2). Give the estimate to two decimal places.

5.[2p] The Mauna Loa annual means for 2014 to 2018, rounded, were 399, 401, 404, 407 and 409 ppm. With x counting years from 2014, what is the slope of the least squares line, in ppm per year?

6.[2p] For the same data, what is the intercept c of the least squares line y=mx+c? Give it to one decimal place.

7.[3p] Find the absolute maximum of f(x,y)=x2-4x+y2-2y on the square 0≤x≤3, 0≤y≤3.

8.[1p] Match each function to the behaviour of its critical point at the origin.

9.[3p] Which of these statements are true?

1.[2p]
The function $f (x, y) = x^{3} - 3 x + y^{2} - 4 y$ has a critical point at $(1, 2)$ . What is $D = f_{x x} f_{y y} - f_{x y}^{2}$ there?

2.[2p]
The same $f (x, y) = x^{3} - 3 x + y^{2} - 4 y$ has exactly one local minimum. What is its value?

3.[2p]
At the origin, $f (x, y) = x^{2} + 6 x y + 4 y^{2}$ has gradient $𝟎$ , with $f_{x x} = 2$ and $f_{y y} = 8$ . What kind of critical point is it?

4.[3p]
Use the second order Taylor polynomial of $f (x, y) = e^{x} \sin y$ at the origin to estimate $f (0.1, 0.2)$ . Give the estimate to two decimal places.

5.[2p]
The Mauna Loa annual means for 2014 to 2018, rounded, were $399$ , $401$ , $404$ , $407$ and $409$ ppm. With $x$ counting years from 2014, what is the slope of the least squares line, in ppm per year?

6.[2p]
For the same data, what is the intercept $c$ of the least squares line $y = m x + c$ ? Give it to one decimal place.

7.[3p]
Find the absolute maximum of $f (x, y) = x^{2} - 4 x + y^{2} - 2 y$ on the square $0 \leq x \leq 3$ , $0 \leq y \leq 3$ .

8.[1p]
Match each function to the behaviour of its critical point at the origin.

9.[3p]
Which of these statements are true?