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The key to finding a functions relative maxima and minima i s the fact that the line tangent to the function at these points is ho rizontal. By finding the x-values which make the derivative equal zer o, we also find the x-values of the relative maxima and minima. These values must be evaluated to find the corresponding y-values. We must also evaluate the second derivative at these x-values. This will sho w us if these points are relative maxima or minima.\n\011Finally we mu st evaluate our function at the x-values of the endpoints of the domai n. We compare the endpoint values to the relative maxima or minima va lues and choose the best result.\n\nSteps to Solve Maximum and Minimum Values\n 1) Find the function which best describes the problem. If necessary, restrict the domain to fit the situation. Use Maple to dr aw the graph. \n2)\011Find the first derivative, set it equal to zero \+ and solve for x. (Note: there may be more than one x-value)\n3)\011Su bstitute these x-values into the function to find the corresponding y- values.\n4)\011Find the second derivative and substitute the x-values \+ from step 2. If the result is positive we have a relative minimum, if the result is negative we have a relative maximum.\n5)\011Finally eva luate function at the endpoints of the domain. Compare these values w ith the relative maxima or minima.\n\nYou may also review these steps \+ using the Calc1_Maxima and Minima project." }}{PARA 0 "" 0 "" {TEXT 286 33 "\011Now solve each of the problems. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 268 0 "" }{TEXT 259 0 "" }{TEXT 260 8 "Problems" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 261 0 "" }{TEXT 262 9 "Problem 1" }}{PARA 4 "" 0 "" {TEXT 271 0 "" }{TEXT 270 156 " A manufacturer wants to design an open box having a square base and a s urface area of 108 sq. in. What dimensions will produce a box with ma ximum volume?" }{TEXT 272 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 263 0 "" } {TEXT 264 9 "Problem 2" }{TEXT 269 1 "\n" }{TEXT 273 33 "Find the poin ts on the graph of " }{TEXT 278 1 "y" }{TEXT 279 4 " = " }{TEXT 275 2 "- " }{XPPEDIT 257 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT 274 41 "+ 4 that are closest to the point (0, 2)." }{TEXT 276 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 258 "" 0 "" {TEXT 265 9 "Problem 3" }}{PARA 259 "" 0 "" {TEXT -1 1 "\n" }{TEXT 277 240 "A rect angular page is to contain 24 sq. in of print. The margins at the to p and bottom of the pate are each 1.5 in. The margins on each side ar e 1 in. What should the dimensions of the page be so that the least a mount of paper is used.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 280 9 "Problem 4" }}{PARA 0 "" 0 " " {TEXT -1 251 " Two posts, one 12 feet high and the other 28 feet hig h, stand 30 feet apart. they are to be stayed by two wires, attached \+ to a single stake, running from ground level to the top of each post. \+ Where should the stake be placed to use the least wire?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 266 1 " " }{TEXT 281 9 "Problem 5" }{TEXT -1 0 "" }} {PARA 260 "" 0 "" {TEXT -1 192 "Four feet of wire is to be used to for m a square and a circle. How much of the wire should be used for the \+ square and how much should be used for the circle to enclose the maxim um total area?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT 267 1 "\n" }}}}{MARK "2 1" 7 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }