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Find the critical numbers of the function.

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

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

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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'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'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'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's the answer to this from.

Universidad Central de Venezuela