Good day students welcome to mathgotserved.com in this group were going to be going over how to find extrema using the first derivative test let's go ahead and take a look at the instructions for the examples that were going to be doing Om we are to find and describe describe the local extrema okay another word for local extrema are you really to extrema okay so this right find and describe the local extrema going talk about extrema are we looking at the Needham on the maximum okay so it to find and describe the local extrema using the first derivative tests okay there is another Om test that can be used to find a describe your extrema known as the second derivative test Om will going to be take a look about in on Leith representations in this will it is little focus on the use of the first derivative test considering how the sign of the first derivative impact the behavior of the function in then using that's two identify or locate our relative extrema the function on the consideration is FO backs equals 5/3 X to the third power -20 X now in this problem we are asked to find the extrema local extrema anytime you see a problem involving extrema you want to use a mnemonic device to help you recall X the procedures involved in the problem-solving process so anytime you deal with extrema you want to remember CVS okay like the store that you go to CVS now Om what's the meaning of CVS CVS basically tell you that when you are finding the extrema you start by looking for the critical values okay critical values and then you also close either endpoints is comes to play when you are given and involved in wish you are to find the extrema the local extrema and then after you find your critical values and you identified endpoints you then going to create is sign chart that the help to use the first derivative tests to describe the behavior of the function using the sign of the first derivative in a you also close either what happens around the endpoints okay isolate go ahead and use a CVS process for this problem now one thing you want to note is that no Interpol is provided here so what are the endpoints there are not okay so for this problem we just little for the critical values in the use Joseph find to create our sign chart and then proceed from there so let's go ahead and start so step number one is what step one is to find the critical values okay find the critical values now how do you find the critical values you find the critical values by finding the where Om the derivative where F prime of zero and when F prime of X is equal to zero or F prime of X does not exists okay when F prime of X equals zero we have the horizontal tangent line situation whenever from does not exist we either have a corner equals the vertical tangents Aomori discontinuity okay so if you think about extrema mostly most likely corners are costs artisan values where you have an extreme in other cases it's a little bit Psalm complicated okay now let's all go ahead and find the first derivative of the function so the function is five 5/3 X to the third so 5/3 X to the third minus swing TX okay so will start by finding the derivative so F prime of X will going to be applying the power rule differentiating these two terms to have 5/3 times you powered is exponent down and then you Om subtract one from it minus the derivative of 20 X is just 20 Valeri Shiga steps so the power here is one you power that down in any subtract one from okay so let's simplify this equation right here this function so will have Om the threes dividing out and we have 5X to the three minus one is to -20 now we'll have X recent is the right power which is just one so 20 times one is just 20 the we now have our first derivative not the funny of first derivative it's beneficial to express it of a fraction okay in this case we have a polynomial Valeri expression of the fraction all divided by what's by one so we're going to find out where F prime of X is equal to zero and where does not exist the last think about this for second will all talk about that later this to for F prime of X equals zero first okay so whenever you looking for F prime of X equals zero and you have your rational expression is express this of the fraction to illustrate that point you the vertically extract the numerator components of your first derivative and you said it equal to zero okay so you set the numerator equal to zero because when the numerator have the value of zero then the denominator then the entire function has the value of zero assuming that the denominator does the have a value zero at that point okay so let's do that you take 5X squarve a quadratic equation here and within solve this by isolating X using the square root property so let's start all we can out 20 to both sides so you have 5X square equals 20 divide both sides by Subscribe Here http://goo.gl/2XXaLS

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