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The graph of a cubic function is shown above. Using the pcints identified write the equation of the cubic function in the for yzar'tbrtbtc)Identify any extrem...

Question

The graph of a cubic function is shown above. Using the pcints identified write the equation of the cubic function in the for yzar'tbrtbtc)Identify any extrema on the graph below .Local MinimaAbsolute MinimumLocal MaximaAtsolute Maximum

The graph of a cubic function is shown above. Using the pcints identified write the equation of the cubic function in the for yzar'tbrtbtc) Identify any extrema on the graph below . Local Minima Absolute Minimum Local Maxima Atsolute Maximum



Answers

Graph the polynomial, and determine how many local maxima and minima it has. $$y=x^{3}-x^{2}-x$$

So if we want to sketch this point when you remember, the first thing we have to do is factor to find the zero. So if we have we fact around X here, what we're left with is X squared minus X minus one. And so, if we factor this, remember, we want to numbers that that multiply too to negative one, but add up to also negative one. Well, we can use the quadratic formula here so we can say we have negative visa one plus or minus the square root of of one squares of one minus four times one times negative one. And so this is over too. So we get one plus or minus one plus or minus the square root of five. This is one plus four one plus or minus the square to 5/2 and so so now we know are two zeros. So this would be one plus square to 5/2 and one minus square to five. Order to, and we also have we also have X is equal to We also have access equal to zero. So this is either our first set, and I remember all of these have a All of these have a multiplicity of one, and so so we can look at our end behavior now. So if X approaches infinity, we can look at our leading term execute and say that our polynomial, because of the odd exponents, are polynomial will also equal positivity. And if x approaches negative infinity than our polynomial will approach negative and 50. And so now we can sketch this graph. Now it comes guest this graph. So if this is axe and this is why well at X is equal to zero we have a zero and let's say this is This is the square root of the square to five is approximately and is approximately greater than greater than two. And so if we have one minus the square to five Well, this is somewhere here. So let's say this is one minus the square to 5/2. And let's say this is this is one plus square defined. So this is, let's say this is one. Then one plus square two finals or two should be somewhere right here. So we know we have two points here and we said that as X approaches as X approaches. Negative infinity. So does our polynomial. So we're going to cross the X axis here and then in and go and go back down. We're gonna go back down here, and and we have, uh We're going to go back up here. We're going to go to infinity as X approaches infinity. We're gonna go up here and keep increasing. And so this is a sketch of our of our graph. And so notice. Here we have We have one local maximum and one local minimum. And so so these are the two local minimums and maximums.

So in this question I have a poly normal given to me by Why's he called two X squared minus to hold you. So if I can fact tries, it already does. X minus toe. That's the roto all killed in tow Express route to whole Cube. So the roads are mine. Esto and my natural plant Bless wrote to So if I observed the graphical becoming like this there's it XX is this is the way access So it will be coming this way. Now this route X equals Ruto ar minus Roto is repeated three times. So I put it That's next access and it will go down like this. Touch a taxi called zero again Raise up gender It's concurrently like this and then again it will keep going this way. So this one drew this minus roto comma zero and this other oneness Ruto comma zero on this point is zero comma minus it. So I have to tell that it has only one bind off minimal and no maximum. So that is how it is. I hope it is clear. Thank you

So since this polynomial is already factored, we can figure out the X intercepts by setting X setting X. That's a X squared minus two to be equal to zero. So we get X is equal to plus or minus the square root of two. And so those are are those. There are X intercepts, but we also know that our end behavior as X approaches infinity. Since this would be X to the six power if we were to expand this the leading term to be X to the six power well, then are Why would also approach infinity and as X approaches negative 50 because of the even exponents, Why would approach infinity and so weaken sketch this now and say, This is our X axis and this c. R y axis. Let's say this is this is positive route to and this is this is let's say this is negative or two. And so if we plug in zero here, what we would get is is zero minus two. So we would have negative, too, to the third power, and this gives us negative eight. So this gives us negative eight. So let's say this is negative eight here. So we have these three points and we said that as X approaches negative infinity. Well, our wire approaches positive feelings. So we're gonna cross the X axis here, and we're gonna go to negative eight, and then we're gonna go upward. Then we're gonna go upward. And so notice here that we have, we have one local minimum, one local minimum and no local maximums, So one local minimum and no local next.

So in this question, I have a degree polynomial given to me. So in orderto brought a graph, I will have the graph something like this. So in this graph, I can observe that the roads are somewhere here. This is one drop. This is, uh, X equal to zero is clearly the rude that is directly visible, But other roads are quite complicated. So these are the roads. If I talk about the points off Maxima so I can see that there is one point off maximum over here. Okay, so I let one Maxima on. If I talk about the minimum, it will be this one minimal. And this is another of anywhere. So I like to many more points. So that's autograph. Has it properties? I hope it's clear. Thank you.


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