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1WDacK OI this, , method., , ILLUSTRATIVE ExAMPLES, 2.() Find the maxim um value, when x +y +2 =k, of*yz,, (i) Find the maximum and minimum, , of x+ y+ z', when, a+by + cz* 1 and lx + my + nz values, =0. Also, interpret the result, geometrically., (ii) A rectangular box open at the top has a, volume of 32 c.c. Find the, dimensions of the box requiring least material, for its construction., (iv) Prove that the rectangular solid of maximum volume which can be, =, , inscribed in a sphere is a cube., (u) Find the volume of the greatest rectangular, parallelopiped inscribed, , 2, , =1, , in the ellipsoid, , Solution. (i) Let, , fx, y, z) r'yi*', , where, , .1), , =, , and, , gl,y,z) =x +y +2 -k, , Also, let, , [RTU 2008], =, , 0, , .(2), , Flx, y, 2) =fla, y, z) + Agt,y,z), , a is the, , Lagrange's multiplier., Fla, y, 2) x*'yiz + a (r +y +z- k), On differentiating (3), partially w.r.t.x, y, z, we get, =, , do, , px, , yz+ A,, , .(3)

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TEXTBOK, , A, , 5.18, OF, oy, , = qy-r'z, , +, , OF, , ENGINEERING, , ., , THEMAT, , 1,, , OF = r2-x'y" + n, , = 0 p x ' yr+ . =0, , Now,, , f+1 =0,, , ox, , =OF, , OF, , and, , 0, , qy-1 x'z + ) = 0 >, , 0 =rz-1x'ya+, , f+, , =0, , =0=;f+i-0, using ean, , On solving equations (4), (5) and (6), we get, , =- Py=- ,2=-4, A, , Now, eqn. (2), k =* +y +2, , k -(p +q +r), , usingeqn, , P+g+r)f, Thus,, , pk, , qk, , rk, , (p +q + r), , (p +q+r), , (p+q+r), , Hence, maximum value, , using egn, , of r"yz, , p'q', p+q+r)P+9+r, , (ii) Let, , fx, y, 2) (x* + y* +z),, =, , glx, y, 2) = ax* + by2 + cz2- 1 = 0, , and, , hlx, y, 2) = lx + my + nz = 0, , Also, let Fx, y, z) = ffx, y, z) + a glx, y, z)+ a hlx, y, 2), , where, , and, , 1, are the Lagrange's multipliers., Fx, y, z) = (r2+y* + z*) + A lax + by? + cz- 1), +, , On, , r + my + nz), , differentiating (4) partially w.r.t. x, y, 2, we get, C 2x + 2, ax *, Ox, , 2y+23by, Oy, , \, , +m

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MAXIMA AND MINIMA OF, , A, , FUNCTION, , F, , and, , 22, , F, , Now,, , =, , 0, , =, , 0, , OF, , y, , OF TWO OR MORE, , On, , 5.19, , +2 Ac7+, , 2x +2, 2y, , 0, , and, , VARIABLES, , +, , ax+, , 2a, , ...(5), , a =0,, , ....6), , by + ,m =0, , 22 +2, , cz + n, , ...7), , 0, , and then, multiplying equations (5), (6) and (7) byx,y and z respectively, , adding. we get, , 2, , (x+y+2)+21, (ar? + by2 +cz?) +, , A, (lx, , +, , +, , my, , hz), , =, , 0, , using eqns. (1),, , 2f+2, = 0, , (2) and (3)], ..8), , A-f, using eqn., , (5),, , (8) in eqn., , get, , we, , 2r-2fax+2=0, .9), 2 f a - 1), , ... (10), , Agm, , Similarly,, , 20fb-1), , ...(11), , 2fc-1), , and, , using equations (9), (10), , and (11) into eqn. (3), we get, , 20f6- 1 ) " 2 f c - 1 ), , 2(fa -1), , 2, , m, (f-1), , fa-1), which, , provides the, , =0, , m, , (fe -1), , maximum, , (, , of, and minimum value, , Ax, y,z) =r+y*, , +z*, , considered as point, Hereplx,y, z) is easily, 1 and plane passing, conicoid ar* + by2 + cz*, between central, -0¥, , Geometrical Interpretation., , of intersection, , through the origin lx, , +, , my, , + nz, , =, , 0 and op', , =, , =, , (x, , -, , 0¥ +, , (y 0¥ +, -, , r+y+z' =fx,y,2), minimum values, , the maximum and, Thus, here we obtained, conicoid and the, intersection of central, , of the point of, , 0), , 0, , z, , ofthe distance, , plane from the origin., , the rectangular box and let S is its surface, (ii) Let x, y, z are the edges of, With given volume V, thien, .1), V=xy2 = 32 (given)

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0, , =, , =0, , y, , 0, , =F, , +, , AXy, , y+ 2z + yz = 0,, , +, , x =, , =-, , 4, , 4, , get, , a --1, rectangular box., , Note: Above problem can also be solved by the method discussed in Remart, , a, , x+y+z=#, , is the, , Px, y, z), , =, , =, , 8xyz, , +, , a (x*, , +, , y* + z, , -, , Fa, y, 2) V+ a (x* +y* +z*-a), Lagrange's multiplier, , sphére of radius a centred at origin., Also, let, , which is, , Subject to the condition, , V= 8xyz, , a), , positive octant and let V denotes its volume, then, ..., , (w) Let x, y, z are the co-ordinates of a vertex of the rectangular soolid i, , (2) of 5.4, , where, , .., , 4, , MATHEMATE,, , Hence, minimum surface is obtained with 4, cm, 4 cm, 2 cm as dimensions, for the given, , the, , 32, , x = 4, y = 4,2 = 2, , 2, , =0, = +2=2 +2, , we, , 2y+ 2x+ àxy, , using equations (7), (8) and (9) into eqn. (1), we get, , Thus,, , 32), , s, , G, ENGINEERINa, , Lagrange multiplio, , (xyz, , x+ 2z + 2x =0, , 2x, , a + 22 + 2x*, , =, , or= 2y, , y + 22 + Ay2,, , =, , solving equations (4), (5) and (6),, , and, , and, , F x . y, 2) = (xy + 2y2 + 22x)+, , where, , OF, , 1s the, the Lagrango', , TEXTBoOK, , =av + 2y2 + 22xx, , A, , On differentiating (3) partially w.r.t. X and y, we get, , Now,, , On, , S, Also, let Fx. v, z) = S+ V, , and, , and, , 5.20

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O= 0, , 0, , 0, , 4zx + y, , 4yz+, x, , =0, , =0,, , ., , ...5), , ..(4), , =, , y, , =, , z, , = ay = ^z, , cube., , Fla,y, 2) =8xyz +, , A, , is the Lagrange's multiplier., , 0, , 4yz+, , =, Now, , 22, , 2y, , oF8xy, +2, 0z, , 82x, and, , c, , Cx, , C=8yz+ 2x, , =0,, , On differentiating(3) partially w.r.t. x,y, 2, we get, , where, , Also, let Fx, y, 2) = V+, , 6, , 2, , Subject to the condition, , V 8xyz, , ..(1), , .4), , .3, , ...(2), , (u) Let x, y, z are the co-ordinates of a vertex of the rectangular, parallelopiped in the positive octant and let V denotes its volume, then, , is a, , Hence, rectangular solid of maximum volume which can be inscribed in a, , X, , -4xyz = 2, , ., , 5.21, , ..(6), 4xy+ az =0, On multiplying equations (4), (5) and (6) by x, y and z respectively,we get, , sphere, , and, , Oy, , =, , 8xy8xy + 22, , 8zx+, +2y, dy, , OF, , =8yz+ +2 x,, , differentiating (3) partially w.r.t. x, y and z, we get, , Now,, , and, , On, , A XIMA AND MINIMA OF A FUNCTION OF TWO OR MORE VARIABLES

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A TEXTB0OK OF ENGINEERING MA, , G MATHEMATIOA, , 5.22, , O=0, , 42x +, , =0, , 4xy+, , dy, OF, , and, , dz, , On multiplying equations (4),, , =0, , (6) and (6) by x, y, 2 respectivel., , we get, , A2, , ., , -4xyz, , =k, (let), 2, C, , using equation (2, , k+ +k=1, 3k = 1, , Hence, maximum volume, , =, , V=, , 806c, , 33, , EXERCISE 5.2, 1. Find the extreme value ofu =r2+y +z2 when, , ar+by+cz?+ 2fy2 +2gzx + 2hxy =1, 2. Find the maximum and minimum values of, , Ax,y,2)= (a*+6y2+c)when x+y+z= 1and l +my +nz =0, 3. Awire oflength d is cut into two parts which are bent in the form, of a square and, circle respectively. Find the least value of the sum of the areas so found, , wi, Lagrange's method of multipliers., Prove that of all rectangular parallelopiped of the same volume, the cube has, the least surface., , 6. Show that the point within a triangle for which the sum of the squares «, its perpendicular distances from the sides is, least in the centre of the coue, , circle., , ANSWERSsF, h, , 1, , 8, , hb, , f, , 0, , C, , m, , f-), 3., , 2, , 4(T+4), , *, , 2, , (f-B f -, , =0, , providesrequired max. an