Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x, y) = 8 sin(xy), (0, 9)

The maximum rate of change of f at (0, 9) is 72 and the direction of the vector is  

Step-by-step explanation:

Given that:

F(x,y) = 8 sin (xy) at (0,9)

The maximum rate of change f(x,y) occurs in the direction of gradient of f which can be estimated as follows;

We can conclude that the  maximum rate of change of f at (0, 9) is 72 and the direction of the vector is  

3. parallelogram

step-by-step explanation:

we turn each of our sides into direction vectors

ab=b-a=(3 - 7, 1 - 0) = (-4, 1)

bc=c-b=(1,6)

cd=d-c=(4,-1)

da=a-d=(-1,-6)

we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram.   we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2.   so it's not a square or a rhombus.  

we check the dot product of direction vectors of adjacent sides.   we have

ab dot bc = -4(1) + 1(6) = 2

that's not zero so the sides are not perpendicular.   we've ruled out rectangle.

the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.

answer: hey there,

the answer you are looking for is a..

hope this you!