in this video we're going to focus on identifying the local maximum and minimum values of the function so let's say if we have a generic shape here we have a local maximum and at this point it's a local minimum so at these extreme values you're always going to have a horizontal tangent line so the derivative is going to equal zero at that point so let's say if we have this function f of x and it's equal to x squared minus four x in order to identify the location of the local maximum and minimum values also known as the relative maximum and minimum values we need to find the first derivative set it equal to 0 and solve for x and that's going to give us the critical numbers and then we could use a sign chart to determine if we have a local max or a local min at that particular critical number so let's go ahead and begin so let's start with the first derivative the derivative of x squared is two x and the derivative of x is one so we have two x minus four and let's set that equal to zero so if we add 4 to both sides 2x will equal 4 and then if we divide by 2 this will give us the first critical number x is equal to 2 since f prime of 2 is equal to 0. so 2 is the critical number now what you want to do at this point is you want to make a number line and place the critical number in the number line now i'm going to write the factored form of the first derivative and that is two times x minus two so let's say if we were to plug in a number greater than two like three will the first derivative be positive or negative three minus two is positive one so it's going to be positive if we plug in a number less than two like one one minus two is negative so if the derivative changes from negative to positive do we have a local max or a local min when the first derivative is negative the function is decreasing when it's positive it's increasing so therefore this is the shape of a local min so we have a local minimum at x equals 2. now to actually find the local minimum value we need to find the y-coordinate of the function so let's evaluate f of two so that's going to be two squared minus four times two two squared is four four times two is eight four minus eight is negative four so the local minimum value is negative four if you want to write your answer as an ordered pair it's going to be two comma negative four so the local minimum is located at x equals two and the minimum value is negative 4. now let's try another problem so let's say that f of x is equal to 2x cubed plus 3x squared minus 12x go ahead and identify all of the relative extrema in this example so let's begin by finding the first derivative the derivative of x cubed is 3x squared and the derivative of x squared is 2x and the derivative of x is one so this is equal to 6x squared plus 6x minus 12. and we need to set the first derivative equal to zero now let's factor first let's take out the gcf which is six six x squared divided by six is x squared six x divided by six is x or 1x negative 12 divided by 6 is negative 2. so now we need to factor this trinomial when the leading coefficient is 1. so we need to find two numbers that multiply to the constant term negative two but add to the middle coefficient one so this is going to be negative two and one negative two times one is negative two but negative two plus 1 is and this should be positive 2 negative 1. i'll take that back because 2 and negative 1 adds up to a positive 1 but 2 times negative 1 is negative 2. so when we factor it's going to be x plus 2 times x minus 1. now we need to set each factor equal to 0. so x plus 2 is equal to 0 and x minus 1 is equal to 0. so we have negative 2 and 1 as the critical numbers of the function so now at this point we need to create a number line so we're going to place negative 2 to the left of 1 and we need to use the factor form of the first derivative to produce the sign chart so let's pick a number that's greater than one let's try two two plus two is positive two minus one is also positive a positive number times a positive number will give us a positive result now there's something that i want you to pay attention to notice that the exponents of each factor also known as the multiplicity is one whenever the multiplicity is an odd number the sign will change if it's even it will remain the same so for this particular critical number we have an odd multiplicity therefore this should be negative it's going to change from positive to negative and for this factor it's also odd so it's going to change from negative to positive and we could test it so if we plug in a number between negative two and one like zero we should get a negative result for the first derivative zero plus two is positive and zero minus one is negative a positive number times a negative number will give us a negative result now let's confirm this one as well so let's try negative three negative three plus two that's negative one and negative three minus one is negative four two negative numbers multiplied to each other will give us a positive result now which critical number is the relative maximum and which one is the relative minimum so negative two is that a relative max or a relative min well the slope changes from positive to negative so on the left side of negative 2 the function is increasing and on the right side it's decreasing so that is a relative maximum so we have a maximum at negative 2. now for the next one the slope is negative so it's the function is decreasing and then the slope is positive now it's increasing so at 1 we have a relative minimum now the last thing that you may need to do is you may need to express your answer as an ordered pair so let's do that for this example but you don't always have to do it for these types of problems it depends on what the problem is asking for so let's determine y when x is one so it's going to be two times one raised to the third power plus three times one squared minus twelve times one so that's gonna be two plus three minus twelve two plus three is five five minus twelve is negative seven so we have the ordered pair one negative seven now for the second one let's replace x with negative two negative two to the third power is negative eight negative two squared is positive four and negative twelve times negative two is twenty-four now two times negative eight is negative sixteen three times four is twelve and negative sixteen plus twelve that's negative four negative four plus twenty four is twenty so we have a local max at negative two comma twenty and so that's it for this problem so now you know how to identify the ordered pairs for the local maximum and the local minimum and we can clearly see that 20 is a lot higher than negative 7. now let's work on one final problem so let's say that f of x is three x to the fourth power minus sixteen x cubed plus twenty four x squared so go ahead and identify the location of any relative extreme values you don't have to find the y coordinates for this problem just the x coordinates so let's start by determining the first derivative so the derivative of x to the fourth is 4x cubed and the derivative of x cubed is 3x squared and the derivative of x squared is 2x so the first derivative is going to be 3 times 4 which is 12 16 times 3 that's 48 24 times 2 is 48 now let's set the first derivative equal to zero and now let's take out the gcf the greatest common factor is going to be 12x 12x cubed divided by 12x that's going to be x squared and negative 48 x squared divided by 12x that's going to be negative 4x and 48 x divided by 12x that's going to be four so now what two numbers multiply to four but add to negative four so this is going to be negative two and negative two negative two plus negative two is negative four but negative two times negative two is positive four so we have x minus two times x minus two so we can rewrite this as 12x times x minus 2 squared now let's set each factor equal to zero so if we set 12x equal to zero and if we divide both sides by 12 x is zero and if we set x minus two squared equal to zero and take the square root of both sides we're gonna have x minus two is equal to the square root of zero is zero and if we add two x is two so the critical numbers are zero and two so now let's make the sign chart let's pick a number that's greater than two let's try three twelve times three is positive three minus two is positive now notice that we have an odd multiplicity for 12x but an even multiplicity for x minus two and when it's even the sign will remain the same but when it's odd it's gonna change so it's gonna change across zero but stay the same across two now let's confirm it so let's pick a number between zero and two let's try one twelve times one is positive one minus two is negative but once you square it it becomes positive so we get a positive result now let's try a negative one 12 times negative 1 is negative negative 1 minus 2 is negative 3 when you square it becomes positive 9 so you have a positive number times a negative number and that will give you a negative result so now the last thing we need to determine is which one is the maximum and which one is a minimum now going from zero to two it's a decrease and then it's increasing so we have a minimum at zero now at two it doesn't change sign so it's neither a minimum or a maximum so the function is probably increasing and then increasing again or it could be increasing this way and then increasing again that way either case none of these represents a minimum or maximum so you can have this shape or you could have that shape so all we have is a minimum at x equals zero you