3 Added third section
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No.

"If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. You suffer a –6 penalty with your regular attack or attacks with your primary hand and a –10 penalty to the attack with your off hand when you fight this way." PFSRD. Notice that it doesn't say "If you choose to make an extra attack, you suffer a penalty". You suffer the penalty if you are wielding the second weapon, i.e. holding it ready to use in an AoO. Think of the penalty as representing how you can't throw your whole body into the rapier thrust if you also want to keep your footing for the whip attack.

Unless you don't need your Move action

Pathfinder is very happy to let you change what you're holding in your hands during your turn. By way of comparison, if you're wielding a quarterstaff two-handed, you can switch to wielding it one-handed while you cast a spell, then switch back before the end of your turn, since both are free actions (search for "two-handed" in my PFSRD link). In your case, it's a Move action to draw a weapon, or to pick it up from the ground (and picking it up from the ground provokes an AoO of its own). It's another Move action to sheathe a weapon, and a free action to drop it.

So in the first round of combat, if you start next to an enemy, you can attack them without wielding the whip, then draw your whip as a Move, ready to make AoOs. Next turn, you have two options: you can sheathe your whip as a Move, then make an attack with no penalty, but not be able to make AoOs after this turn, although next turn you can. Alternatively, you can drop your whip as a free action, attack, then pick up your whip this turn, and attract an AoO of your own.

Edit: It's ambiguous

The rules say "When you fight this way". It's not really clear whether "fight this way" means "wielding two weapons" or "making an extra attack". On the one hand, the penalty for two-handed fighting doesn't affect your AoO, so it would make some sense if it was the latter. On the other hand, you're still using your whip for the purposes of fighting if you're wielding it without making an extra attack. The Move action to start "wielding" a weapon might not just mean taking it off your belt, it might also mean adjusting your footing so you're ready to attack with the whip. Honestly, if I was DMing this, I'd rule wielding was fighting, just because it gives you more options to consider during a combat (when do you draw your whip? If you pick up your whip off the ground and draw an AoO, does that let someone else do something cool without being attacked?) But YMMV, and in particular, YDMMMV.

No.

"If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. You suffer a –6 penalty with your regular attack or attacks with your primary hand and a –10 penalty to the attack with your off hand when you fight this way." PFSRD. Notice that it doesn't say "If you choose to make an extra attack, you suffer a penalty". You suffer the penalty if you are wielding the second weapon, i.e. holding it ready to use in an AoO. Think of the penalty as representing how you can't throw your whole body into the rapier thrust if you also want to keep your footing for the whip attack.

Unless you don't need your Move action

Pathfinder is very happy to let you change what you're holding in your hands during your turn. By way of comparison, if you're wielding a quarterstaff two-handed, you can switch to wielding it one-handed while you cast a spell, then switch back before the end of your turn, since both are free actions (search for "two-handed" in my PFSRD link). In your case, it's a Move action to draw a weapon, or to pick it up from the ground (and picking it up from the ground provokes an AoO of its own). It's another Move action to sheathe a weapon, and a free action to drop it.

So in the first round of combat, if you start next to an enemy, you can attack them without wielding the whip, then draw your whip as a Move, ready to make AoOs. Next turn, you have two options: you can sheathe your whip as a Move, then make an attack with no penalty, but not be able to make AoOs after this turn, although next turn you can. Alternatively, you can drop your whip as a free action, attack, then pick up your whip this turn, and attract an AoO of your own.

No.

"If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. You suffer a –6 penalty with your regular attack or attacks with your primary hand and a –10 penalty to the attack with your off hand when you fight this way." PFSRD. Notice that it doesn't say "If you choose to make an extra attack, you suffer a penalty". You suffer the penalty if you are wielding the second weapon, i.e. holding it ready to use in an AoO. Think of the penalty as representing how you can't throw your whole body into the rapier thrust if you also want to keep your footing for the whip attack.

Unless you don't need your Move action

Pathfinder is very happy to let you change what you're holding in your hands during your turn. By way of comparison, if you're wielding a quarterstaff two-handed, you can switch to wielding it one-handed while you cast a spell, then switch back before the end of your turn, since both are free actions (search for "two-handed" in my PFSRD link). In your case, it's a Move action to draw a weapon, or to pick it up from the ground (and picking it up from the ground provokes an AoO of its own). It's another Move action to sheathe a weapon, and a free action to drop it.

So in the first round of combat, if you start next to an enemy, you can attack them without wielding the whip, then draw your whip as a Move, ready to make AoOs. Next turn, you have two options: you can sheathe your whip as a Move, then make an attack with no penalty, but not be able to make AoOs after this turn, although next turn you can. Alternatively, you can drop your whip as a free action, attack, then pick up your whip this turn, and attract an AoO of your own.

Edit: It's ambiguous

The rules say "When you fight this way". It's not really clear whether "fight this way" means "wielding two weapons" or "making an extra attack". On the one hand, the penalty for two-handed fighting doesn't affect your AoO, so it would make some sense if it was the latter. On the other hand, you're still using your whip for the purposes of fighting if you're wielding it without making an extra attack. The Move action to start "wielding" a weapon might not just mean taking it off your belt, it might also mean adjusting your footing so you're ready to attack with the whip. Honestly, if I was DMing this, I'd rule wielding was fighting, just because it gives you more options to consider during a combat (when do you draw your whip? If you pick up your whip off the ground and draw an AoO, does that let someone else do something cool without being attacked?) But YMMV, and in particular, YDMMMV.

2 Added second section
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No.

"If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. You suffer a –6 penalty with your regular attack or attacks with your primary hand and a –10 penalty to the attack with your off hand when you fight this way." PFSRD. Notice that it doesn't say "If you choose to make an extra attack, you suffer a penalty". You suffer the penalty if you are wielding the second weapon, i.e. holding it ready to use in an AoO. Think of the penalty as representing how you can't throw your whole body into the rapier thrust if you also want to keep your footing for the whip attack.

Unless you don't need your Move action

Pathfinder is very happy to let you change what you're holding in your hands during your turn. By way of comparison, if you're wielding a quarterstaff two-handed, you can switch to wielding it one-handed while you cast a spell, then switch back before the end of your turn, since both are free actions (search for "two-handed" in my PFSRD link). In your case, it's a Move action to draw a weapon, or to pick it up from the ground (and picking it up from the ground provokes an AoO of its own). It's another Move action to sheathe a weapon, and a free action to drop it.

So in the first round of combat, if you start next to an enemy, you can attack them without wielding the whip, then draw your whip as a Move, ready to make AoOs. Next turn, you have two options: you can sheathe your whip as a Move, then make an attack with no penalty, but not be able to make AoOs after this turn, although next turn you can. Alternatively, you can drop your whip as a free action, attack, then pick up your whip this turn, and attract an AoO of your own.

No.

"If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. You suffer a –6 penalty with your regular attack or attacks with your primary hand and a –10 penalty to the attack with your off hand when you fight this way." PFSRD. Notice that it doesn't say "If you choose to make an extra attack, you suffer a penalty". You suffer the penalty if you are wielding the second weapon, i.e. holding it ready to use in an AoO. Think of the penalty as representing how you can't throw your whole body into the rapier thrust if you also want to keep your footing for the whip attack.

No.

"If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. You suffer a –6 penalty with your regular attack or attacks with your primary hand and a –10 penalty to the attack with your off hand when you fight this way." PFSRD. Notice that it doesn't say "If you choose to make an extra attack, you suffer a penalty". You suffer the penalty if you are wielding the second weapon, i.e. holding it ready to use in an AoO. Think of the penalty as representing how you can't throw your whole body into the rapier thrust if you also want to keep your footing for the whip attack.

Unless you don't need your Move action

Pathfinder is very happy to let you change what you're holding in your hands during your turn. By way of comparison, if you're wielding a quarterstaff two-handed, you can switch to wielding it one-handed while you cast a spell, then switch back before the end of your turn, since both are free actions (search for "two-handed" in my PFSRD link). In your case, it's a Move action to draw a weapon, or to pick it up from the ground (and picking it up from the ground provokes an AoO of its own). It's another Move action to sheathe a weapon, and a free action to drop it.

So in the first round of combat, if you start next to an enemy, you can attack them without wielding the whip, then draw your whip as a Move, ready to make AoOs. Next turn, you have two options: you can sheathe your whip as a Move, then make an attack with no penalty, but not be able to make AoOs after this turn, although next turn you can. Alternatively, you can drop your whip as a free action, attack, then pick up your whip this turn, and attract an AoO of your own.

1
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No.

"If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. You suffer a –6 penalty with your regular attack or attacks with your primary hand and a –10 penalty to the attack with your off hand when you fight this way." PFSRD. Notice that it doesn't say "If you choose to make an extra attack, you suffer a penalty". You suffer the penalty if you are wielding the second weapon, i.e. holding it ready to use in an AoO. Think of the penalty as representing how you can't throw your whole body into the rapier thrust if you also want to keep your footing for the whip attack.