# Consider Fixed-Point Iteration with g(x) = x + x 3 . (a) Show that x = 0 is the only fixed point….

Consider Fixed-Point Iteration with g(x) = x + x3. (a) Show that x = 0 is the only fixed point. (b) Show that if 0 <>0 < 1,="" then="">0 <>1 <>2 <……….. (c)="" show="" that="" fpi="" fails="" to="" converge="" to="" a="" fixed="" point,="" while="" g’(0)="1." together="" with="" exercise="" 27,="" this="" shows="" that="" fpi="" may="" converge="" to="" a="" fixed="" point="" r="" or="" diverge="" from="" r="" when="" |g’(r)|="">

Exercise 27

Consider Fixed-Point Iteration with g(x) = x − x3. (a) Show that x = 0 is the only fixed point. (b) Show that if 0 <>0 < 1,="" then="">0 > x1 > x2 … > 0. (c) Show that FPI converges to r = 0, while g (0) = 1. (Hint: Use the fact that every bounded monotonic sequence converges to a limit.)

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