Find the critical numbers of the function.
$ f(x) = 2x^3 - 3x^2 - 36x $
$-2$ and $3$
we will find the critical numbers of the function F of x equals two X cube minus three X squared minus 36 X. So we remember that the numbers see in the domain of the function of To give a function search that derivative at that point does not exist Or exist and is equal to zero. Are called the critical numbers numbers in the domain of the function Where the derivative death not exist or existed in the support zero. So this is our definition here. So the first thing we have to notice here is that the domain of the function is of this function here is a real numbers because simple in normal, so you can read three. So it is well defined everywhere in the real numbers. So it's so many all the set of all the real numbers. Yeah. Following that We find the derivative formula to function and is six X squared minus six X minus 36. We know a polynomial has preservatives everywhere. In fact, is given by this formula, which is fine at all. Real numbers. So the derivative of path exists and every real number X. It means that we are not going to find critical points of this function that satisfies this condition of they're already not defined are not existent. So this part is not taking into account. So we are looking for points in the domain Where the first serve a tip is zero because of the many serial numbers. Then we've got to find numbers real numbers or a real number or real numbers. Where did the relative is Syria? Yeah. Then the critical numbers of they give them function. Yeah. Our does real numbers for which Jax first derivative of the function is zero. Yeah. So we start by equating to 02 1st derivatives. That will give us the critical point. So six x square. It's the same thing as saying that six X sq -6 X -36 is zero. And that's the same as saying that six times x square minus x minus six is zero. Because we can factor out this is six from the formula. And we can see that If this expression here is zero the whole expression here will be zero because six factor does not make any difference. The whole freshen being equal zero depends only at this paranormal here because six is different from zero. So this equivalent say that X square minus X -60. Mhm. And now this falling on the here can be factored out as X um Plus two times X -3 equals zero. That is because if we develop this product here, we get this polynomial or what is the same. We have found two numbers 2 and 83 Oh is the product of those numbers is equal to the independent terms 96. And some of the two numbers these the coefficient of X. And with that we factor out pornography and now it is to see that the roots of these paranormal is and then of these other anomalies or are X equals -2 and X equals three. Here it or because any of those various notify the equation. So we are going we are looking at the equivalence. But at the end We say that the two regal points that are negative two and 3 good book because both of them fulfill the condition of being in the domain of the function because they are simply too real numbers And the derivative is equal to zero. So then the critical okay. Numbers of a given function. Yeah. Are negative two and three. And disease the answer to these. Pro