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Question Number 175785 by otchereabdullai@gmail.com last updated on 06/Sep/22

Commented by Ar Brandon last updated on 07/Sep/22

#include <stdio.h> #include <math.h> int main(void) { double fx, gx, xn, x_m; printf("Enter initial value:"); scanf("%lf", &xn); for(int i=0; i<100; i++) { fx = pow(xn, log(2)/log(3))-sqrt(xn)-1; gx = log(2)/log(3)*pow(xn,log(2)/log(3)-1) -0.5/sqrt(xn); x_m = xn - fx / gx; xn = x_m; } printf("x ≈ %.2f\n", xn); return 0; }

Commented by Ar Brandon last updated on 07/Sep/22

$$x=9$$$$\mathrm{Newton}-{\mathrm{Raphson}}^{\prime }\mathrm{s}\mathrm{method}\mathrm{with}\mathrm{initial}$$$$\mathrm{value}{x}_0=1\mathrm{and}100\mathrm{repetitions}.$$

Commented by Frix last updated on 07/Sep/22

$$\mathrm{this}\mathrm{is}\mathrm{again}\mathrm{no}\mathrm{math}\mathrm{example}\mathrm{but}\mathrm{a}\mathrm{magic}$$$$\mathrm{trick}\mathrm{example}.\mathrm{obviously}\left(\mathrm{at}\mathrm{least}\mathrm{for}\mathrm{me}\right)$$$$x=9.\mathrm{but}\mathrm{no}\mathrm{exact}\mathrm{method}\mathrm{is}\mathrm{possible}.\mathrm{or}$$$$\mathrm{else}\mathrm{try}$$$$x^{{\log }_{2}3}=\sqrt{x}+1$$

Answered by Ar Brandon last updated on 07/Sep/22

Answered by LordKazuma last updated on 07/Sep/22

$$x^{{log}_{3}2}=\sqrt{x}+1$$$$letx=3^t\Rightarrow \sqrt{x}=3^{\frac{t}{2}}$$$$3^{t\cdot {log}_{3}2}=3^{\frac{t}{2}}+1$$$$3^{{log}_3{2}^t}=3^{\frac{t}{2}}+1$$$$2^t=3^{\frac{t}{2}}+1$$$$\mathrm{the}\mathrm{value}\mathrm{of}t\mathrm{that}\mathrm{statisfies}\mathrm{this}\mathrm{equation}\mathrm{only}$$$$t=2\Rightarrow x=3^t\Rightarrow x=9$$

Commented by LordKazuma last updated on 07/Sep/22

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