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Good day students welcome to mathgotserved.com in this clip were going to be going over how to find critical values of the function rights it instructions for the examples us follows we are to for number one find the critical values for the function F of X equals 3X minus 2X to the to third okay so before we get started let's go over what critical values are quick a right soul all like is recall the fact that Om are to go values critical values are when the derivative F prime is equal to zero or the derivative F prime of X does not exist okay so these are the two cases where we have critical values now when the derivative is equal to zero what's we have easy horizontal tangent line okay we have a horizontal tangent line when the derivative is equal to zero now that is Om one of the 06 wishes we have a critical value now the second cases when the derivative does not exist what results in the nonexistence of the derivative or basically when is a function not differentiable will in that case you have either a vertical turn their line or you have no tangent line at all okay so when you have critical values and easy way to remember is when you have or 10) note K horizontal tangent for when the derivative is equal to zero vertical tangents or no tangent line window derivative does not exist right so let's keep those appoints in mind now let's take a look at the steps involved in finding the critical values of a function number one you have to first find the derivative of the given function F prime of X next you will then find when Om the derivative is equal to zero and when it does not exist okay so fine X where we have the fully equations F prime of X equals zero horizontal tangent line or F prime of X does not exist our right let's go ahead and apply this step two this problem right here so we have F of X the original function equals 3X minus 2X to the to third power go ahead and differentiate using the power rule F prime of X is going to be the derivative of 3X which is three minus the derivative of all 2X to the to third is to times to third X recent the to third minus one okay let's simplify that we have 3-4/3 X to the one can be written as three over three so he subtract one or three over three from to third you be left with negative one third okay now Om let's skills simplify the little bit further we can write this as 3-4 over 3X to the one third using the reciprocal properties of exponents to reciprocate the term with the negative exponent now what we looking for horizontal tangent or whether derivative is nonexistent we want to try to express the function as a quotient of two expressions okay in this case we have a function here in an integer here this is not the best Om method to help us accurately determining the Om X values where we have zero tangent or nonexistent tangent so what I'm going to do is proceed to unite these two terms into one so will have one clear numerator and one clear denominator okay so how do we combine these two all express this integer have a fraction the LCD of one and 3X to go one third is 3X to the one third so multiply this by 3X to the one third of the top and the bottom okay and that will give us 3089 multiply the three's together so will have 9X to the one third over 3X to the one third -4 over 3X to the one third now can I combine these two fractions absolutely because we have identical denominators so F prime of X can be written as 9X to the one third -4 divided by Om 3X to the one third now writing it in this format is excellent because the have a clear numerator and a clear denominator by can help us find when the derivative is equal to zero or nonexistent okay so let's go ahead and do it let's start with where F prime of X is equal to zero in this case you just simply extract the numerator of the reduced function of we are going to extract the numerator and set it equal to zero our right so we have 9X to the one third -4 so that equal to zero we might ask why did we do that's well this is a reason if you said this function equal to zero and you cross multiply what happens is that the denominator gets multiplied by zero and in memory to gets multiplied by one so you always be left with the numerator okay so whenever the numerator of the rational function in reduced form is equal to zero entire function attain the value of zero is longest Psalm double impact the denominator value okay so let's see solve this for X to do that's will first at four to both sides this is just busy all the right now and we have 9X to the one third equals for the can up proceed to divide both sides by nine divide by nine divide by nine and have X to the one third equals 49 and then to get X isolated we will raise both sides of the equation to the third power that is that inverse of the third root okay on the left side we have X on the right side we

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