1.

Consider the following.

f(x,y) = 3x²+y²6x2y+ 4

Find the point (x,y) wheref(x,y) has a possible relative maximum or minimum. (Give the exact point.)

(x,y) =

,

2.

Consider the following.

f(x,y) = 2x²3y³16x+ 36y+ 32

Find the points (x,y) wheref(x,y) has a possible relative maximum or minimum. (Give the exact points.)

(x,y) =

,

(point with smallery-value)

(x,y) =

,

(point with largery-value)

3. Consider the following.

f(x,y) = 6xy²2x³3y⁴

Find all the first and second partial derivatives andD(x,y).

f

x

=

f

y

=

²f

x²

=

²f

y²

=

²f

xy

=

D(x,y) =

Both first partial derivatives of the functionf(x,y) are zero at the points below. Use the second-derivative test to determine the nature off(x,y) at each point. If the second-derivative test is inconclusive, so state.

(a) (0, 0) relative maxrelative min neither max nor mininconclusive

(b) (1, 1) relative maxrelative min neither max nor mininconclusive

(c) (1,1) relative maxrelative min neither max nor mininconclusive

4. Consider the following.

f(x,y) =x²2xy+ 3y²+ 4x12y+ 35

Find the exact point (x,y) wheref(x,y) has a possible relative maximum or minimum. Then use the second-derivative test to determine, if possible, the nature off(x,y) at this point. If the second-derivative test is inconclusive, so state.

(x,y) =

,

---Select--- relative max relative min neither max nor min inconclusive

5. Consider the following.

f(x,y) =x²+ 4xy+ 2y⁴

Find the exact points (x,y) wheref(x,y) has a possible relative maximum or minimum. Then use the second-derivative test to determine, if possible, the nature off(x,y) at each of these points. If the second-derivative test is inconclusive, so state.

(x,y) =

,

---Select--- relative max relative min neither max nor min inconclusive (smallestx-value)

(x,y) =

,

---Select--- relative max relative min neither max nor min inconclusive

(x,y) =

,

---Select--- relative max relative min neither max nor min inconclusive (largestx-value)