 In this video, we provide the solution to question number one for practice exam number four for math 1210 In which case we're asked which of the following are the critical numbers for the function f of x equals x cube minus 2x squared minus 7x plus 3 Now remember the critical numbers are exactly those numbers which make the first derivative equal to zero or make the first derivative Not exist. So D and E there means does not exist now given that the function is a polynomial The derivative will likewise be a polynomial and so there are no points for it doesn't exist So we need to figure out where the derivatives equal to zero so accomplish that we're gonna compute the derivative just by the usual derivative rules Power rule and such so we get the derivative. It's gonna be 3x squared minus 4x minus 7 We set that equal to zero And so we could use the quadratic it's a quadratic function that we have to solve right here this equation So we can use the quadratic formula because we could complete the square We could try to factor it. I'm a fan of factoring. So if that's possible I want to give it a try so we have to look for number the numbers three times negative seven So three times negative seven that's equal to negative 21 Are there factors of negative 21 that add to negative four and I'm just gonna use the numbers that already in front of me Notice if I just take three minus seven that is negative four So I can factor this thing by groups You can try to do a little bit of a guess and check if you want to Or you can go through the fact in my group process I'm gonna try to be a little bit quick here. So I think I can guess my way through You're gonna take 3x times an x. That's the only way you can get a 3x squared has to be a 3x times x to get a negative seven Well, it's gonna be a one time seven one has to be negative one has to be positive And so I don't want to do something like a negative seven over here because that would give me a 3x times negative so that's like a negative 21x that's not right So it looks like what I need is to get a plus one right here in a negative seven right here So again, you can kind of guess your way through it and so notice if we try this out here 3x times x is 3x squared negative seven times one is a negative seven Then if we take 3x times one, that's 3x and negative seven times x is a negative seven x That's gonna have to be negative four x. So that is the right factorization We set that equal to zero and so we're gonna get that the first one of by the zero product property If 3x minus seven equals zero that means x equals seven thirds And if x plus one equals zero that means x equals negative one for which then we see that the correct answer is D