Question

For each of the Claims, decide whether it is True or False. If it is True, briefly explain why it is true. If it is False, provide a counter-example to the Claim.

Answer 1

. Solutim: n let f: IR→IR be class c² if cab) 6.18? is saddle point of_f, then Hf(0,b) is indefinite. . This statement is true let cab) is a saddle point of t Then RI - $? <o where Refexta, b)_ Ta fyy tab_ s- fxy (a, b) = fya le, by mainly hered four possibility for RT -s² co (i) PEREO I=o0 ; 5230 Then Hf(a,b) - Fxxka,b) Excyl(a,b). I fyxl cab) fyylla,62 | Hilab) = R 5l ISI Now ai (principal minor) = R=0 - 1 = RT-S2 2o hence Hf(a,b) is indefinite Co if R=0 TF0;520 then from 0 & H fab) = 10 si IsI -doh =-6220

bence HF (ab) is indefinite Ci_B20 T=0 5230 iz from 040 HElab) = R S T in. I ts Oil DER so aa 1,30 - = -5?co-> 09. hence Hf (ab is indefinite civo can R28 TEO e s²so that RT :-$2co. such Hela b) = Rs ISTI - =Rco Ag = RT-5220 con Da 20. - - hence Hflab) is indefinite. hence H f(a,b) ir indefinite matrix
Thanks and happy learning