13.8 Day 2 Extrema of Functions of Two Variables Finding Absolute Extrema
Find the absolute extrema of Graph of Use number sense to find the solution Problem 46 Find the critical points and test for the relative extrema. List the critical points for which the Second Partials test fails. Solution to problem 46 fx = 3x2 -12x + 12 = 0 fy = 3y2 + 18y + 27 = 0
Solve and get x=2 and y = -3 At (2,3) fxx fyy (fxy)2 = 0 Test is inconclusive. Look at graph on next slide Graph of 46 Problem 52 Find the critical points of the function and from the form of function, determine whether a relative maximum or minimum occurs at each point. Solution to problem 52 fx = -2x(y-1)2(z+2)2 = 0
fy = -2x2(y-1)(z+2)2 = 0 fz = -2x(y-2)2(z+2) = 0 Solving these yields critical points (0,a,b) (c,1,d)(e,f,-2) By number sense we obtain that these are maximums Problem 54 Find the absolute extrema of the function over the region R. The region R is the region in the xy-plane with vertices (2,0), (0,1) and (1,2) Solution to 54 fx = 4(2x-y) = 0 when 2x=y fy = -2(2x-y) = 0 when 2x=y
Boundary: (2,0), (0,1) and (1,2) fx = 4(2x-y) = 0 when 2x=y fy = -2(2x-y) = 0 when 2x=y By inspection we can see that this is a minimum (z=0) We also need to check the boundaries The boundaries are y = x x1 y= -1/2x+1 0x2 y =-2x x2 Plug in boundary points and equations to look for critical points Graph of 54 Homeworkp odd Answers on the next slide Movie Answers 1. 7 Pounds 2. Vertical Limit 3. Speed 4. American Pie 5. xXx or Cube (both work) 6. Lord of the Rings 7. WALL-E Hours 9. The Elephant Man 10. *batteries not included (note: the * is in the movie's name)