Question

For the function F(x) = 24 – 14x² + 60.x2 – 702 find minimum value using two methods - a. Newton's method starting with initial point of 1 b. Golden section in the interval [0,2] required tolerance =0.001

Answer 1

Given, F(x) = x'- 14x} + 60x?- 70x as Newton's method. No. 1 F'(x) = 4x3 - 42 x?+ 120 x - 70 Newton's method formula is X n + 1 en fran f(x) Iteration 1) no x 1 x. f(xol p'ex) (134, 14 (1,37 6011)?- 70x1 41113-4261,?+ 1 + 120(1) -70 7.9166 Iteration 12: n=1 X2 f(x) X 1 f (x) (7.91661"- 14(2. 14(2.9166)+6012.916632-70 2.9166 2.9166 462.916673_428.9166)?+ 120(3.9166) - 70 1.4934 Iteration (3): n ? f(x2) хз x2 #'(x2) (1.4934)4-14(1.49342+6011.4934)’- 70x1.49.34 -1.4934. 4(1.4934).*. 42(1.4934)? +140 (1.4934) - 70 : 1,9223 Iteration 4 n= 3 X4 - x₂ 3 f'(x3) (1.9223) -34(1.9223)+60(1.9223) ?- 70 (1.9223) 4 (1.9223)3 - 4 21 1.9223)+12011.9223) - 70 -1.9223- 1.8821 Iteration 5 n: 4 x 5 = C4 f(xu) I'(xu) (1.88219 141.8321'46041.88217? 70(1.8821) 4(1.5821) 3 - 4211.55211? + 120(1.8821) 1.8821 70 X y = 1.88 205 using Newton's method is Hence, minimum rawe 1.88205 205 golden section method d 0.618034°(b-a) x1-ard K2=b-d check-il(abs(x2-x1).<0.001 than 1 else f(x) = x^4-1473 + 60x^2 - 70% Keration a b 1 0 0 0 0 2 3 0 4 0 5 0 6 7 ld x2 t(x1) check 2 1.236068 1.236068 0.763932 18.95816015 -24.3606798 1.236068 0.76393205 0.76393205 0.47213595 24.3606798 21.0985144 1.236068 0.402135981 0.9427193 0.763932019 23.59246166 -24.3606798 0.94427193 0.291 796089 0.763932038 0.652475842 -24.36067979| 23.8370353 0.9M27193 0.180339904 0.832815746 0.763932027 24.28788706 -24.3606798 0.832815746 0.111456192 0.763932034 0.721359553 -24.36067979 -24.2579475 0.832815746 0.068883716 0.79024327 0.763932029 24.36690716 -24.3606798 0.832815746 0.012572479 0.806504508 0.790243267 24.34952629 -24.3669072 0.806504508 0.026311239 0.790243269 0.780193269 -24.36690716 24.3695868 0.790243269 0.01626124 0.78019327 0.773982028 24.36958684 -24.3681278 0.790243269 0.010049999 0.784032028 0.780193269 -24.36929608 -24.3695868 0.784032028 0.006211241 0.78019327 0.777820786 24.36958684 -24.3693117 0.784032028 0.003838758 0.781659545 0.780193269 -24.36958301-24.3695868 0.781659545 0.002372489 0.780193269 0.779287061 -24.36958684 -24.3695228 0 0 0.47213595 0.47213595 0.652415842 0.652475142 0.721359553 0.763932029 0.763932029 0.763932029 0.773982028 0.773982028 0.777820786 0.777820786 8 0 9 10 0 0 11 12 0 0 13 0 14 1 hence, x= (x1+x2)/2=0.779740165 Scanned with CamScanner

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