Question

3. Find the following:

a.

Suppose

f_xx = − 24x , f_yy = − 4 , f_xy = f_yx = 3

and the critical points for function f are

A = (−0.5585, −0.4189) and B = (0.7460, 0.5595)

Find the value for D for each critical point and then classify the critical point using the second derivative test.

a) D(−0.5585, −0.4189) = −44.6160 ; saddle point at (−0.5585, −0.4189) ; D(0.7460, 0.5595) = 80.6160 ; relative maximum at (0.7460, 0.5595)

b) D(−0.5585, −0.4189) = −62.6160 ; relative maximum at (−0.5585, −0.4189) ; D(0.7460, 0.5595) = 62.6160 ; saddle point at (0.7460, 0.5595)

c) D(−0.5585, −0.4189) = −62.6160 ; saddle point at (−0.5585, −0.4189) ; D(0.7460, 0.5595) = 62.6160 ; relative minimum at (0.7460, 0.5595)

d) D(−0.5585, −0.4189) = −62.6160 ; saddle point at (−0.5585, −0.4189) ; D(0.7460, 0.5595) = 62.6160 ; relative maximum at (0.7460, 0.5595)

e) D(−0.5585, −0.4189) = −44.6160 ; saddle point at (−0.5585, −0.4189) ; D(0.7460, 0.5595) = 62.6160 ; relative maximum at (0.7460, 0.5595)

f) None of the above

b.

Suppose that

f ( x , y ) = 4x ³ − 3xy + 2y ² ,

(0.1875 , 0.1406) is a critical point,

f _xx | _{(0.1875 , 0.1406)} = 4.5000 , and

D (0.1875 , 0.1406) = 9 .

Which of these statements describes the graph of f at (0.1875 , 0.1406) ?

a) f has a relative minimum value at f (0.1875 , 0.1406) = −0.0132.

b) f has a saddle point at f (0.1875 , 0.1406) = −0.0132.

c) f has a relative maximum value at f (0.1875 , 0.1406) = −0.0132.

d) f has a relative minimum value at f (0.1875 , 0.1406) = 0.0659.

e) f has a relative maximum value at f (0.1875 , 0.1406) = 0.0659.

f) f has a saddle point at f (0.1875 , 0.1406) = 0.0659.

g) None of the above