Question

Let f(x, y) = −x 3 + y 2 . Show that (0, 0) is a saddle point. Note that you cannot use the second derivative test for this function. Hint: Find the curve of intersection of the graph of f with the xz-plane.

Answer #1

(3)If H(x, y) = x^2 y^4 + x^4 y^2 + 3x^2 y^2 + 1, show that H(x,
y) ≥ 0 for all (x, y). Hint: find the minimum value of H.
(4) Let f(x, y) = (y − x^2 ) (y − 2x^2 ). Show that the origin
is a critical point for f which is a saddle point, even though on
any line through the origin, f has a local minimum at (0, 0)

Let f(x) = x*(2-x) if x>=0, or x*(x+2) if x<0
i) graph the function from x=-3 to x=+3. If you like
WolframAlpha, use
Piecewise[{{x*(2-x),x=>0},{x*(x+2),x<0}}]
If you like Desmos, use f(x)= {x>=0:x*(2-x),
x<0:x*(x+2)}
(for some reason, when you paste that it, it forgets the first
curly-brace { so you’ll need to add it in by hand) Or, you can use
this, but it makes it less clear how to take the derivative:
f(x) = -sign(x)*x*(x - 2*sign(x) )
ii) Find and...

Let y = x 2 + 3 be a curve in the plane.
(a) Give a vector-valued function ~r(t) for the curve y = x 2 +
3.
(b) Find the curvature (κ) of ~r(t) at the point (0, 3). [Hint:
do not try to find the entire function for κ and then plug in t =
0. Instead, find |~v(0)| and dT~ dt (0) so that κ(0) = 1 |~v(0)|
dT~ dt (0) .]
(c) Find the center and...

Let f(x,y) = e-x^2 + 5y^2 - y. Use the
Second Partials Test to determine which of the following is
true.
A) f(x,y) has a saddle point at (0, 1/10)
B) f(x,y) has a relative minimum at (0, 1/10)
C) f(x,y) has a relative maximum at (0, 10)
D) f(x,y) does not have a critical point at (0, 1/10)

Problem 1.
(1 point)
Find the critical point of the function
f(x,y)=−(6x+y2+ln(|x+y|))f(x,y)=−(6x+y2+ln(|x+y|)).
c=?
Use the Second Derivative Test to determine whether it is
A. a local minimum
B. a local maximum
C. test fails
D. a saddle point

Find the point on the curve y = sqrt(x) is closest to the point (3,
0) , and find the value of this minimum distance.
Use the Second Derivative Test to show that this value is a
minimum. Show work

Consider function f(x, y) = x 2 + y 2 − 2xy and the 3D graph z =
x 2 + y 2 − 2xy. (a) Sketch the level sets f(x, y) = c for c = 0,
1, 2, 3 on the same axes. (b) Sketch the section of this graph for
y = 0 (i.e., the slice in the xz-plane). (c) Sketch the 3D
graph.

The function
f ( x , y ) = x 3 + 27 x y 2 − 27 x
has partial derivatives given by
f x = 3 x 2 + 27 y 2 − 27,
f y = 54 x y,
f x x = 6 x,
f y y = 54 x,
f x y = 54 y, and
f y x = 54 y,
AND has as a critical point. (You need NOT check
this.)
Use the second...

Let f(x, y) = (2y-x^2)(y-2x^2) a. Show that f(x, y) has a
stationary point at (0, 0) and calculate the discriminant at this
point. b. Show that along any line through the origin, f(x, y) has
a local minimum at (0, 0)

Let F ( x , y ) = 〈 e^x + y^2 − 3 , − e ^(− y) + 2 x y + 4 y 〉.
a) Determine if F ( x , y ) is a conservative vector field and, if
so, find a potential function for it. b) Calculate ∫ C F ⋅ d r
where C is the curve parameterized by r ( t ) = 〈 2 t , 4 t + sin
π...

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