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Solutions to Problems; Chapter 7 363 5.1. The Lagrange polynomial of degree not exceeding n - 1 which assumes the values f(ui) a ...
364 Answers to Exercises and Solutions to Problems as required. 5.4. Let c, m, n be fixed, and define g(x) = (x+c)ā. Then f(m) = ...
Solutions to Problems; Chapter 7 365 For case (3), it suffices to consider functions g(z) whose minimum value is -1 and maximum ...
366 Answers to Exercises and Solutions to Problems 5.6. We deal first with the quadratic situation. Let f(x) = (x - v)(x - u) = ...
Solutions to Problems; Chapter 7 367 which contradicts the induction hypothesis. 5.7. Since only differences of the ni are invol ...
368 Answers to Exercises and Solutions to Problems and f(1) = -p-q 1 -(1/2)(&l) e-p 5 -q+(1/2)(Jz-1) 5 (&i-1)-1 < 0. ...
Solutions to Problems; Chapter 8 369 Since x2 5 x, it follows that 1 - x 5 dn. Hence --t + (1/2)(Jz+ 1) - (l/2)(& - 1) I J1- ...
(^370) Answers to Exercises and Solutions to Problems 24 = (x + y + 2)ā - (x3 + y3 + z3) = 3Cx2y + 6xyz j 8 = x(y + z)~ + (x2 + ...
Solutions to Problems; Chapter (^8 371) Second solution. Let y = z + u. Then (3u + 2)x2 + (3u2 + 2u)x + (uā + 8) = 0. The discri ...
372 Answers to Exercises and Solutions to Problems We can now solve for x and y. 8(x + ,/jj) = 8u3 = (zā + 3%~) + (3z2 + w)& ...
Solutions to Problems; Chapter (^8 373) Try w = 1. Then we wish to find v and z such that 12v2 - 23 = ,r2. Two obvious solutions ...
374 Answers to Exercises and Solutions to Problems (c) First solution. The equation is equivalent to 8(z2 + XY + y2 + 1) = (x + ...
Solutions to Problems; Chapter 8 375 (iii)x+y+%=l;y=z.Theny=z=$(l-x)andx2+2y2=zimply 0=3x2--442+1=(32-1)(x - 1). Hence, (x, y, Z ...
376 Answers to Exercises and Solutions to Problems Second solution. Since p(x) and q(x) are continuous, then p(x) > q(x) for ...
Solutions to Problems; Chapter (^8 377) First solution. Let f(Y) = (~-fi-mY-\/z+mY+~-~)(y+Jz+~) = Y - lOy2 + 1. Then f(x-45) = ...
(^378) Answers to Exercises and Solutions to Problems Thus, from (b), if p < -l/2, there is no solution. If p < -l/4, then ...
Solutions to Problems; Chapter (^8 379) When p = -l/4, then a = b. Hence there is a unique solution if and only ifp=-1/4orO<p ...
380 Answers to Exercises and Solutions to Problems asinu+bsinv+csinw=O. Shifting two terms to the right side in each equation, s ...
Solutions to Problems; Chapter (^8 381) Since this guarantees that either 1 is a zero or f(1) and f(-1) have oppo- site signs, h ...
382 Answers to Exercises and Solutions to Problems The system p+q+r=o up+vq+wr =0 xp+yq+%r=o (*I in the variables p, q, r has ...
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