If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:4:53

let K equals four so that f of X is equal to one over x squared minus four X determine whether F has a relative minimum a relative maximum or neither at x equals two justify your answer all right well if f of X is equal to this then F prime of X they gave us F prime of X in terms of K so f prime of X is going to be equal to in this case K is 4 so it's going to be 4 minus 2 x 4 minus 2 x over x squared minus 4 x squared minus 4x minus 4 x squared so now we know f of X and we know F prime of X and if we were looking for relative minima or relative maxima we would be interested in in points and the critical points and especially where F prime of X is equal to 0 and so if we said f prime of X is equal to 0 well we could say well when does this numerator equal a 0 and so you could say when does 4 minus 2x equal 0 you had 2x to both sides you get 4 is equal to 2 X or X is equal to 2 X is equal to 2 and they told us that or I guess we've confirmed that at at F prime F prime of 2 does indeed equal to 0 so this is definitely an interesting point so let's think about whether before when X is less than 2 is f prime of X increasing or decreasing and then when X is greater than 2 is f prime of X increasing or decreasing and that that'll let us know if this is a minima or a Maxima so let's say so when when x is less than 2 we could test something out we could say F prime of so let's just say F prime of 1 so if X is less than 2 and you actually don't even have to test F prime of 1 I mean you could if you want you would have 4 minus 2 4 minus 2 times 1 so that's going to be positive and then this down over here is always going to be negative because you have a squared right over here so when X is less than 2 f prime of X is greater than 0 and you could try this out with different X's if you like you could say for example for example f prime of 1 is equal to 4 minus 2 which is 2 over 1 minus 4 which is negative 3 but then squared is equal to 2 ninths and then we could say when when X is greater than 2 f prime of X F prime of X well when X is greater than 2 you have 4 minus 2 times something larger than 4 so this is going to be negative so f prime of X is going to be less than 0 appear is going to be negative down here is not going to be negative and so f prime of X is going to be less than 0 so if if we are increasing as we approach something and then our slope is 0 and then we are decreasing well then this is going to be a maximum point so that means that we have a maximum maximum point point at X is equal at X is equal to or we could say yes we have F has a relative maximum relative maximum at x equals at x equals 2 and actually listen to me right X has a relative maximum I'm just use the words they are using relative relative maximum at X is equal to 2 now another way that you could have tried to do it is you could have try to take the second derivative of this and then solve whether that was positive or negative and whether it's concave upwards or downwards but taking the second derivative this it gets quite hairy and anytime you're taking the AP test and you find yourself going down a really really hairy path like taking the derivative of well taking the second derivative derivative of F which would be the first derivative f-prime anytime you see yourself going down a hairy path like that it might work but it's probably not the optimal path so not an easier way to just think okay well what is f prime doing as we approach this or another way is thinking is the function increasing as we approach from below and is it is it increasing or decreasing as we get beyond that point beyond x equals two

AP® is a registered trademark of the College Board, which has not reviewed this resource.