Examples for Section 12.8

EXAMPLES FOR SECTION 13.8

EXAMPLE--Tangent Planes and Max-Min

Find the points on the graph of the function below where the tangent plane is horizontal.

f(x,y) = x³ - 3xy + y² DPGraphPicture (z-axis scale: 1 unit = 10) DPGraphPicture with the 2 horizontal tangent planes

Since the surface can be represented by F(x,y,z) = f(x,y) - z = x³ - 3xy + y² - z = 0 and grad F will give a vector normal to the surface we need to find points where grad F = < 0 , 0 , c > where c is any nonzero constant.

grad F = < f_x , f_y , -1 > so we are looking for points on the surface where f_x = f_y = 0.

f_x = 3x² - 3y = 0

f_y = -3x + 2y = 0

The solutions to this system are (0,0), and (3/2,9/4).

f(0,0) = 0 and f(3/2,9/4) = -27/16 so the points on the surface where the tangent plane is horizontal are (0,0,0) and (3/2,9/4,-27/16). The picture on the left below shows the graph of the surface around the point (0,0,0) and the picture on the right shows the graph around the point (3/2,9/4,-27/16). Click on each graph to see an animation that will include the tangent plane at the critical point on that graph.

To determine the nature of these points where the tangent plane is horizontal we can apply the Second Partials Test.

d = f_xxf_yy - (f_xy)² f_xx = 6x, f_yy = 2, f_xy = -3

d(x,y) = 12x - 9

d(0,0) = -9 so (0,0,0) is a saddle point.

d(3/2,9/4) = 9 and f_xx and f_yy are positive at (3/2,9/4) so (3/2,9/4,-27/16) is a relative minimum point.

On the right is a graph of the surface over the x-interval [-1,3] and y-interval [-1,3]. Click on the picture at the right to see an animation. It would be difficult to visually determine the nature of the critical points from the graph on the right but less difficult from the graphs above. The analytical approach tells us for sure. It is also pretty obvious from this DPGraph Picture with the two critical points (z-axis scale: 1 unit = 1) and this DPGraph Picture of the surface with the two horizontal tangent planes. Use the cursor keys to look at the pictures from different views. Maple Version

Now let's consider the problem of finding the absolute minimum and absolute maximum value of the function over the square region of the xy-coordinate plane described by the x-interval [-1,3] and y-interval [-1,3]. See the picture on the right above or this DPGraph Picture (z-axis scale: 1 unit = 10). The two critical points found above yield -27/16 as a possible minimum and 0 as a possible maximum. We must now evaluate the function on the boundary of the square region being investigated.

Check of the boundary x = -1 with y in [-1,3]

f(-1,y) = g(y) = -1 + 3y + y²

g'(y) = 3 + 2y = 0 if y = -3/2 which is not in [-1,3]

g(-1) = -3 g(3) = 17

Check of the boundary y = -1 with x in [-1,3]

f(x,-1) = j(x) = x³ + 3x + 1

j'(x) = 3x² + 3 cannot equal 0

j(-1) = -3 (as above) j(3) = 37

Check of the boundary x = 3 with y in [-1,3]

f(3,y) = h(y) = 27 - 9y + y²

h'(y) = -9 + 2y = 0 if y = 9/2 which is not in [-1,3]

h(-1) = 37 (as above) h(3) = 9

Check of the boundary y = 3 with x in [-1,3]

f(x,3) = k(x) = x³ - 9x + 9

k'(x) = 3x² - 9 = 0 if x = 3^1/2 and -3^1/2 -3^1/2 is not in [-1,3]

k(3^1/2) = 9 - 6(3^1/2) which is greater than -3

k(-1) and k(3) would put us at corners of the square already evaluated above.

Thus the minimum value of the function over the region is -3 and occurs at (-1,-1,-3)

and the maximum value of the function over the region is 37 and occurs at (3,-1,37).

Here is a DPGraph Picture of the surface

f(x,y) = (x³ - 3xy + y²)/10 and the planes z = -0.3 and z = 3.7

graphed over the square region of the xy-coordinate plane described by the x-interval [-1,3] and y-interval [-1,3].

Section 13.8 #25 DPGraphpicture

DPGraphpicture including the horizontal tangent planes (tilt the graph to see them both clearly) Quicktime movie showing many views

Click on the Maple picture to see a different view. The blue traces intersect at (0,0,0) and the red traces intersect at (1,1,-1).

Section 13.8 #28 DPGraphpicture

Here is another DPGraph Picture (finer mesh). Try looking at z-slices on the scrollbar to roughly approximate the relative maximum and minimum values to support the answers given above. Critical Point Zoom