I have a credit card balance of 10,000+ dollars with an interest rate of 14%. If I cut the balance down to 3000 with one payment, can the credit card company increase my interest rate because they are angry I payed off a large amount so quickly and they want to make up for the lost interest they would have received?
No. That's pretty unlikely. Card issuers typically base your rate on your credit score. Paying down debt reduces your percent of available credit used, and improves your score until you are in the 1-20% range. That's optimum. To this issuer, you are one of a million customers, there's no emotion in this, just numbers to them.
For what it's worth, if a card issuer raises your rate, you are permitted to "not accept" the rate, stop using the card for new charges, and pay at the current rate. Of course this doesn't apply to zero interest deals, only to increases to your regular rate.
Short answer: No, not normally.
Long Answer: It depends on the contract. If the 14% is some sort of special offer, with conditions, then if you violate those conditions, they can jack you up to whatever the 'normal' rate is. But outside of that condition, I can't see any reason why they would wish to penalize you for making a payment. You will note that there is no "maximum" payment on the bill.
Secondly, even if they do jack up the rate to 28%, you're still better off paying $70 on 3000, than you are paying ~120 on 10k. Then tell them where to stick their card and get a new one.
Credit card companies will typically not care about your individual credit card account. Instead they look either at a "package" of card accounts opened at roughly the same time, or of "slices" of cardholder accounts by credit rating. If an entire package's or slice's balance drops significantly, they'll take a look, and will adjust rates accordingly (often they may actually decrease rates as an incentive to increase you use of the card).
Because credit card debt is unstructured debt, the bank cannot impose an "early payment penalty" of any kind (there's no schedule for paying it off, so there's no way to prove that they're missing out on $X in interest because you paid early). Generally, banks don't like CC debt anyway; it's very risky debt, and they often end up writing large balances off for pennies on the dollar. So, when you pay down your balance by a significant amount, the banks breathe a sigh of relief. The real money, the stable money, is in the usage fees; every time you swipe your card, the business who accepted it owes the credit card company 3% of your purchase, and sometimes more.