Question

Let Which of the following are TRUE? Select ALL that apply. Please show all your work.

fr, y) = sin(x + 7/2) +y?

a. has a local maximum at whenever is an even integer

b. has a saddle point at whenever is an even integer

c. has a saddle point at whenever is an odd integer

d. has a local minimum at whenever is an odd integer

Answer 1

given function fix,y) = spoirex + + y2 Solution - frast gweill find coitical point : Now differenciatinig ea 0 « @ partially wit on a partially wirto x sy i sin (rex+5) 7+y2 no COB 1 txt) xer) - op het | Sin (14+$.) +y?] OX for coitical point of to a sf = 0 => IT 205 1t*+ ) = 0 . xy = 6 – COB 111a + b ) = 0 - y = 0] COBI 104 + ][ ) = 60% (n=+) 3 mx + y = 72 thu & nez anani Tx= n nez Then critical point say (K,O) where Kez.

egno ☺ Now from fax = -resin (t* +72). / - fyy 781 foryzo] Then fxx fyy-fay = -m? sin Irun +TE) *9-09 fox fgy -fry = -9 r2 sin (4x+1). at (ko), 2002 fax fyy-fxy = -92 sin ( KIT +!) = -92? sin ( KIT + Elor LV .: For local maxinuen or minimum fox fyy-frey so sin (kt+) should be negatice K=2. Sir ( 911 + 1) = sinn - 1. so f(x,y) has no maxinum or minimum of for arbitrary Integer so option @ is wang. But even integer fryfyy-fry² <o .. 80 reitheo maxiinium rer minimum so opeo) f has saddle pant (ko) when kis evenintage , ;-) option (6) is tous - Iftar k=3, sin ( 3r+ )=-1, so fun= r2>o . K=5 Sin (511+5) = -1 , pat odd intage fara fyy-frey so so option & is abstrue