This turned out to be more complicated than I thought. Here is what I came up with. Your move at 'A' saves the bottom group if white plays correctly and, if black insists, can even cost him the game. White can choose to save the top if he wishes, but either way black kills the other group and white loses.
Black should not try to kill the group white helps. He can win without it and you shouldn't strive to kill groups or complicate the game when you can win in a simpler way. When you are losing then yes, complicate stuff and try to get your opponent to make a mistake or miss something, but when you are winning keep it simple.
(;GM[1]FF[4]CA[UTF-8]AP[CGoban:3]ST[2]
RU[Japanese]SZ[9]KM[0.00]
AW[fb][gc][hd][df][ff][gf][dg]AB[eb][cc][ec][de][fe][ge][he][cf][cg]LB[ef:A]C[How will Black kill one group of White's choosing? It is White's move.]
;B[]LB[gb:B][fh:A]C[Well, from this position white gets to pick. Save group 'A' or save group 'B' - group 'A' is clearly bigger.]
(;W[ef]C[I don't think black can kill white and unless white messes up I think trying is game losing when black already can win.]
(;B[hg]LB[ef:A]C[That move at 'A' makes this one not work so well.]
;W[hf]
;B[if]
;W[ee]
;B[ed]
;W[fd]C[This white cut looks scary for black.]
;B[gg]C[If black still goes for a caputring race...]
;W[gd]
;B[ie]
;W[id]
;B[ig]C[Three liberties to 4, black should win this capturing race.]
;W[eh]C[But White has this move.]
(;B[fh]C[Black can threaten to live here]
(;W[ch]C[And if white lets him live...]
;B[hi]
;W[fg]
;B[fi]
;W[ei]C[Now both groups live for white.]
;B[bh]
;W[bi]
;B[ah]
;W[ci]
;B[fc]
;W[hb]
;B[fa]
;W[gb]
;B[ga]
;W[ha]
;B[ea]TW[ia][ib][hc][ic][eg][dh][di]TB[aa][ba][ca][da][ab][bb][cb][db][ac][bc][dc][ad][bd][cd][dd][ae][be][ce][af][bf][ag][bg][gh][hh][ih][gi][ii]C[But black still wins the game by a lot.])
(;W[fi]C[If white tries to stop him like this it fails.]
;B[gi]C[Black can threaten to take.]
;W[ei]C[So white backs off]
;B[fg]
;W[hh]C[White has to play here to stop black from getting 2 eyes. But...]
;B[dh]
;W[eg]
;B[di]C[Shortage of liberties])
(;W[hi]C[If white goes for the vital point of the shape then things get complicated. I cannot say for sure this is the best variation.]
;B[fi]
;W[ch]
;B[fg]
;W[bh]
;B[bg]
;W[ah]
;B[be]
;W[ei]
;B[hh]
;W[gi]
;B[ii]
;W[ih]LB[dh:A]C[Then we get a ko for life for black. Black has 1 local threat at 'A', so I think he wins the ko.]
;B[dh]
;W[di]
;B[ii]
;W[dd]
(;B[ce]C[If black defends this then I'm not srong enough to say how this goes down.]
;W[ih]
;B[dc]C[I think White has too many threats coming from that stone.]
;W[eg]C[Now the ko is do or die.]
;B[ii]
;W[ea]C[Black must not respond]
;B[gh]
;W[ca]
;B[bb]
;W[ba]
;B[ab]
;W[ag]
;B[af]
;W[fc]
;B[db]
;W[da]
;B[cb]
;W[fa]TW[ga][ha][ia][gb][hb][ib][hc][ic][dh][ai][bi][ci]TB[ac][bc][ad][bd][cd][dd][ae][bf][ih][gi][hi]C[White wins if the komi is at least 1.5])
(;B[gh]C[If black ignores the result seems a bit better.]
;W[ce]
;B[cd]
;W[de]
;B[eg]
;W[bi]C[White has to live now.]
;B[dc]
;W[dh]
;B[bd]
;W[ag]
;B[bf]
;W[af]
;B[ae]
;W[ea]
;B[da]
;W[fa]
;B[fc]
;W[hb]TW[ga][ha][ia][gb][ib][hc][ic][ai][ci]TB[aa][ba][ca][ab][bb][cb][db][ac][bc][ad][ih][gi][hi]C[If the komi is at least 4.5 white actually wins.])))
(;B[ch]C[If black insists on going for the kill right now...]
;W[fg]
;B[dh]
;W[gh]C[Black dies]
;B[hh]
;W[hi]
;B[ei]
;W[ih]))
(;B[hb]C[This simply kills the top right you seem to have read that out.]))
(;W[hb]C[White saves the top right like this. Nice and easy, it is the vital point for both sides.
But then the bottom dies.]
(;B[id]
;W[ic]
;B[ie]
;W[gd]C[At least 2 eyes.])
(;B[hg]
;W[hf]
;B[if]
;W[ee]
;B[ed]C[This is Atari now.]
;W[ef]
;B[gd]
;W[fc]C[This poke helps to solidify white's corner.]
;B[fd]
;W[gg]
;B[hh]
(;W[ch]C[If white tries to get more eyespace this way...]
;B[bh]
;W[dh]
;B[fh]C[Black comes in on this side.]
;W[gh]
;B[gi]
;W[eh]
;B[ei]
;W[fg]
;B[fi]
;W[di]
;B[hi]C[Just one eye for white. He dies.])
(;W[gh]
;B[ch]
;W[dh]
;B[di]C[If white tries to just go for a vital point right now...]
(;W[ei]
;B[fh]SQ[eg][fg][eh][fi]C[This is the vital point of the "Boxy 5" shape.]
;W[ci]
;B[bh]LB[di:A]C[The eye at 'A' is false and will never become real, White dies.])
(;W[fh]
;B[ei]C[He still can't make 2 eyes.]
;W[fi]
;B[hi]
;W[eh]
;B[ci])))))