Question

f(x,y)= (3x^2)+(4y^3)-24xy+29

does this have a local max or min or both? does it have a
critical point?

Answer #1

Find all local maximum or local minimum or saddle point for f(x,y)=
8y^3 + 12x^2 -24xy

f(x)= 12x- x^3
1. where is the local min? x-value
2. where is fhe local max? x-value
3. what is the inflection point? (x and y)
4. for which interval is the graph "concave down"?
for
what interval is the graph concave up?

For f(x)=2+3x+x3 find:
A.increasing and decreasing intervals
B. Local max and min coordinates
C. Concativity Intervals (UP & DOWN INTERVALS)
D. Inflection points coordinates
E. Graph of f(x)

Find the absolute max and min for f(x,y) = (x-3)^2+y^2 on
D={(x,y):0 ≤ x ≤ 4 , x^2 ≤ y ≤ 4x}.

Find the absolute maximum and minimum of f(x,y)= 3x+4y within
the domain x^2+y^2 less than or equal tp 2^2
Find the points on the cone z^2=x^2+y^2 that are closest to the
point (3,4,0)

Let f(x,y) = 3x^2y − 2y^2 − 3x^2 − 8y + 2.
(i) Find the stationary points of f.
(ii) For each stationary point P found in (i), determine whether
f has a local maximum, a local minimum, or a saddle point at P.
Answer:
(i) (0, −2), (2, 1), (−2, 1)
(ii) (0, −2) loc. max, (± 2, 1) saddle points

(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)

. Find the absolute max and min of the following function f(x,
y) = x^2 + 2xy − y^2 − 4x, 0 ≤ x ≤ 2, 0 ≤ y ≤ 2.

Find the relative maximum and minimum values.
f(x,y)=x^3+y^3−24xy

Suppose that f(x,y) = x2−xy+y2−3x+3y with −3≤x,y≤3
1. The critical point of f(x,y) is at (a,b). Then a=
and b=
2.Absolute minimum of f(x,y) is
and absolute maximum is

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