find-the-critical-numbers-of-the-function-fx-x2-e-3x

Question

Find the critical numbers of the function.

$ f(x) = x^2 e^{-3x} $

Chris Trentman · Accepted Answer

were given a function and we&#x27;re asked to find the critical numbers of this function function is F. Of X equals X squared times E. To the negative three X. Now, to find the critical numbers, we want to find where the derivative F. Prime is either zero or undefined. So F prime of X. Buy. The product rule is two X times E. To the negative three X minus three X squared times E. To the negative three X. Now obviously this is defined everywhere. We want to find when this is zero. Well, this is when if we factor this X Times 2 -3 x times E to the negative three X equals zero. Yeah. And therefore is equal to zero when X equals zero or Positive 2/3. And these are the critical numbers of the function.

Deandre Johnson · Answer

right, guys. So for this problem, we&#x27;re ask defying the critical numbers of the function. And we know that critical numbers occur when the derivative of the function is equal to zero. So I think our first step would be to find a bit of a function. I&#x27;m gonna use the product rule, which is essentially the derivative of this first function here, X squared to x times the second function, which is me to the negative three X and then we&#x27;re gonna add the derivative of the second function should be this time&#x27;s the first ones, which is just exquisite. And now we&#x27;re going to set this equal to zero and solve for X thinks again neither. So I think a smart new here would be to factor on decks like so syrup. From here, we can see that X times. This will be zero. So that means that we can say that exit was Europe. And it&#x27;s essentially like we&#x27;re dividing both sides. But X to get rid of this packs here. And then we write this portion and further Saul. Yeah. Cool, huh? And now from here, I think I&#x27;m gonna add this to the other side. So it&#x27;ll be like this. Yeah. And now I think I&#x27;m gonna divide on both sides, but in negative Three X. And now we&#x27;ll have to. This cancer goes three. I&#x27;ve been there. What about books? That an accident? Too green. So now we can see that from here we have X equals zero Mexican Children during these critical numbers of my function.

Erna Bisschoff · Answer

in this example, we have to find the critical values off if X were if X is given by Ex Cute plus X squared minus X now, to find the critical points, we have to sit the derivative off if X equal to zero. So to find the derivative, it will be three x squared plus two x minus one. Putting this equal to zero. We can find X, by fact arising, and it will be three weeks minus one and X plus one for the two factors, which gives us the values off eggs off one over three, the third or X equal to negative one. Right? So these this gives us the X values off the critical points, not necessarily the turning points that can only be proved by the second derivative, which is listen for another day to find the corresponding Why values? We have to substitute the values off this eggs. These two values off X into the original if off its in other words, the y value that corresponds with 1/3 real gate. We&#x27;ll put the third into the original if off X and calculating this the Y value is negative. Five over 27 right? So, in other words, our co ordinates here is 1/3 and negative five over 27. The why value that corresponds off the negative one. It&#x27;s found by plugging in the negative one into the original if off X, which will give us the value off, right. It gives us the value off unless one on this gives us the coordinates off negative one and one for the Wye Valley, Right? If you have a look at the graph, you can see on the growth that there are indeed turning points at point. D and E are using the second derivative taste. We can confirm that, but for now, this is all needed. The coordinates off the turning points off the critical points in this case.

Oswaldo Jiménez · Answer

we want to find the critical numbers of the function F of X equals X to the negative two times natural logarithms of X. So we remember the definition of critical number. We say that a number C. In the domain of the function F. It&#x27;s a critical number of the function. If either the first derivative at that number is zero or first derivative at that number does not exist. So this is the definition of critical number. And then we calculate the derivative. But first we remember here, which is the domain of these functions as we can see, we have this is the same as natural logarithms of X over eggs square. So we have a fraction division of functions. So it&#x27;s important that the denominator is not zero. So X cannot be zero. But also the natural algorithm imposed the condition and eggs to be positive. And in fact if X is positive, it is already different from zero. And so the general condition of the points X to evil they currently dysfunction is that X. B. Positive. So we can say that the main is The Interval zero Plus Infinity. That is the real numbers. So she&#x27;s dad eggs is positive, that&#x27;s the main. And is there in this interval where we look at the curriculum numbers by definition are points or numbers into the main of the function. So now we calculate the first derivative of F. We have a product, we&#x27;re going to use this form instead of this product here. So we apply the proper role of the derivative and we get derivative extra native to is native to X To the negative three times natural rhythm of X Plus X to the -2 times. These are derivative of natural algorithm effects. She&#x27;s one of rex and this is the same as negative, negative to natural rhythm of X over X cube plus. Then he&#x27;s an anti particle putting the denominator, we get one over X cube. So this is the same as 1 -2. natural rhythm of eggs over eggs Q. That is the first relative is equal to one minus two natural rhythm of X over eggs Q. And so let&#x27;s see if there are points in the domain where conservative is not defined. In fact we have again, natural algorithm effects which is a fine for all positive numbers. And the division the fraction is possible When X is not zero. So two conditions again exist. Material X is positive. Give us that X must be positive, which includes already that X is not zero. So uh these relativity fine exist, let&#x27;s say better exist or every point or value eggs into the main of F. That is for every positive value eggs. And that means that we won&#x27;t find critical points with the condition that the derivatives that does not exist. So the only critical point or critical numbers of the given function F are those values those X. Positive in the real numbers, search that the first derivative E. Zero. So we got to solve the equation after the difficult zero for positive values of X. So we start with first serve a difficult zero. It&#x27;s the same saying that 1 -2 natural rhythm of x over X. Q. is zero. And this is the same as saying that 1 -2 natural rhythm effects zero. And that&#x27;s important to remark that this equivalence implies that we are already uh the values of X that are positive. That is any of these two sides of the equivalence imposed this condition because we have the natural liberation in both of them. And so we have this uh equation here which is the same as Natural sodium of eggs equal to 1/2. And that&#x27;s the same as E. To the natural rhythm of X equal to E. To the 1/2 power. And that&#x27;s the same as remember the exponential and the natural logarithms are in their functions of each other. So that gets two eggs Equal Italy 1/2 power in that Square TV. That is the only solution of this equation is a positive number X equal to scores of E. So the only critical number after given function F. Is X equal squirrel of E. So it&#x27;s very important that we uh find first domain of the function they ended the find the derivative. Look at where the a derivative exist if there are points in domain where derivative does not exist. They are critical points or numbers. And besides that we got to find where the derivative is equal to zero. So we have done that and we found only one critical point in this example

Lynnsey Davison · Answer

Hello. So we have this problem here, and we need to find, um, the critical numbers of the function. So our first step is to find of prime of X on equal it 20 to find X values that would make this slope of the function equal to zero. So a prime of X would then be equal to six x squared plus two x plus two. So that&#x27;s the first derivative of F of X. Now we&#x27;re going to equate a prime of X equal to zero, which means that X six x squared plus two X plus two is equal to zero. Um, Now, the second step is to find the X values that would make, um, this truth six x squared plus to expose to equal to zero. What? Um, so normally, you would try Teoh Easiest ways to try the factor that&#x27;s out. But in this case, it&#x27;s not easy to factor this equation out. So we&#x27;re going to use quadratic formula and so called reddit formulas equal to X equals negative B plus or minus two square root. Ah B squared minus for times a time. See over to a um so that means they&#x27;re a it&#x27;s six r b is too, um and R C is, too. Which means that now, if we put these values into the quadratic formula, we get negative two plus or minus the square root of two squared minus hapless, minus four times six times, too over two times six. And then if we solve now, just simplify this. You get minus to plus or minus square root of four, minus 48 over 12. And then that would simplify, too, Um, negative to you Poster minus the square root of negative 44 over 12. So before we simplify further, you can see that inside is irrational is a negative number. So this means that this is an imaginary number, which means that there is no rial solutions for four X. What does that mean For the critical numbers? That means there are no critical numbers. And then So that means that, um, the slope of a function is constantly increasing or decreasing, and in this case, it&#x27;s constantly increasing eso help. This helps

Oswaldo Jiménez · Answer

Yeah. We want to find the critical numbers of the function F of X equal to x cubed plus X squared plus two X. So, first we find the derivative of F. That function is six X squared plus two eggs plus two. And this is relative exist at any ex any real number eggs. The main in fact of the function is the whole real numbers. Because this expression is paranormal. Is it fine for every real number X. So the men of F. S are the real numbers and in a relative exists for every eggs in the real numbers. That is is two facts together implies that the only critical points of function F Are those eggs for which there are TV zero. Mhm. Yeah. That is because remember the definition of critical number of a function. Yeah. All right. The X. Or values in the domain of the function for which the relative does not exist or It exists and is equal to zero. So uh that&#x27;s why we look first at the domain, the function that is the real numbers. And then we see where derivative exists. In this case it exists for every real number. That is for every X. In the domain of the function. Therefore the only critical points or numbers of these functions are those values of X. In the real numbers for which the derivative is equal to see. Mhm. So the equation we gotta solve is this f derivative equal zero. And that&#x27;s equivalent to looking at the formula here, derivative six X square last two eggs Last two equals 0. And that&#x27;s the same thing in two common factory at two times three X square plus eggs plus one. Because two is against a constant different from zero is equivalent to, sorry, here is equal zero. And is going to that the expression inside parenthesis through three X squared plus X plus one Be equal to zero. That that is the equation we get us solved with this one here because sold in this equation we have all the eggs for which the first derivative of the function is equal to zero. So let&#x27;s find that by applying the formula. Or best, we&#x27;re going to calculate the discriminate of this equation because second degree polynomial equals zero. So you see it&#x27;s criminal expression which is the square minus four K. C or B is a division of X to the one a suggestion of X square and sees an independent term. So in this case is equal to one square Wayne is four times 3 times one That is one 12. That is -10. And because this expression is going to appear inside the square root in the general formula, dissolution of the second order and a question, you know that with this condition here, we won&#x27;t have real solutions, then, um There is no number X. And the real numbers such that yeah, three X square plus six plus one equals zero. That is the same as saying that the first derivative of F Different from zero for all eggs, Nah, real numbers. And then taking into account what we said above, we can conclude that this function has no critical points and that the final answer cause this problem.

Chris Trentman · Answer

the problem is finding the critical numbers off a function ofthe lacks a coach&#x27;s ex squire. Tam&#x27;s into negative reacts. First, we compute the derivative of this function of ex problem. This is a coach. You to Lux. I&#x27;m still connective. Reacts us, explain Pam&#x27;s connective reacts hands negative three. So this case, he can&#x27;t chew to axe minus three x Claire Pam&#x27;s into connective. Three acts I&#x27;m. Then we lot derivative equal to zero half to axe minus three. X square is equal to zero self This equation have axe is equal to zero or ex escort you two over three. So the critical numbers off the function are Cyril and to third.

Chris Trentman · Answer

Let&#x27;s use the product rule F one G plus. After you won to take the derivative Septus equals zero. We end up with their first solution of X equals zero. Next set, two months, three X equal to zero. Then we end up with our next solution of X equals to divide by three.

Chris Trentman · Answer

we will find the critical numbers of the function F of X equals X squared times Exponential of -3 X. So remember that C. In the domain of function is a critical number. If to uh any of two things happened. The first river activated point does not exist or it exists but he&#x27;s zero. Mhm. So this is a definition of critical point function. So the first condition is at the point being the domain of function. And after that, if any of these two things happened then the number is critical number. You give a number of F. Of course. Okay, so here we find the first relative of F. In this case we have the product of two functions. So again, product rule is for derivative of the first factory square is two X times the exponential of native three X plus The first factor x squared times the derivative of eat the -3. X. is Each of the -3 x times the derivative of the exponents, which is -3. And we get to eggs eat the negative three eggs -3 x square feet and 93 eggs. And now we can take Common factor out X. E. to the -3 eggs. And we get inside parenthesis to minus three X. That is the first derivative of F. As expression eggs Eat the negative three eggs times 2 -3 eggs. And this derivative here exist for any real number eggs. And we also know that the domain of this function is the real numbers. Because the formula defining F can be evaluated at any real number X. So we have the main of F equal the real numbers and the first derivative exists at any real number X. These two things together imply the stat. We apply that um The only critical numbers of F are those real numbers such that for which theory of the T. V. Zero? Because it cannot happen. The first possibility this one here because if conservative exists everywhere. So the critical points can only be Those values for is derivative zero. But now we start with this equation for conservative of f equals zero. That&#x27;s the same. And saying that this formula here is zero and his ex he into the negative three X times two min minus three eggs equals zero. And this is equivalent. In fact that X equals zero or 2 -3 eggs equals zero. That is because this factor here is not zero for any real number That is we have a product of three functions eggs times this exponential of -3 eggs. And these factor here to -3 eggs. Because the factor exponential, snap the three X can never be zero. It implies that the other two factors can be serious, any of them or both of them. That is or X equals zero, or the factor too many three X is equal zero. And that&#x27;s the same. Are saying that X0 or eggs equal two thirds. If we solve this equation here for X and then these are the two values that for which the preservative is zero, that is. These are the critical point now and the critical numbers of so the only critical numbers of F R zero and two thirds. And that&#x27;s the answer to this from.

Problem

Find the critical numbers of the function. $ f…

Problem 43 Medium Difficulty

Find the critical numbers of the function.

$ f(x) = x^2 e^{-3x} $

Answer

The given function has two critical numbers, $0$ and $\frac{2}{3}$

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The given function has two critical numbers, $0$ and $\frac{2}{3}$

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Integration Review - Intro

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