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Find the intervals of increase and decrease of the functions.$$f(x)=x^{3}-4 x+1$$...

Question

Find the intervals of increase and decrease of the functions.$$f(x)=x^{3}-4 x+1$$

Find the intervals of increase and decrease of the functions. $$f(x)=x^{3}-4 x+1$$



Answers

Determine the interval(s) on which the function is increasing and decreasing.
$$f(x)=4(x+1)^{2}-5$$

Okay, so on this one, it gives us the equation one over X. And it wants to know when it's increasing, decreasing and constant. So if you've washing the other videos, you know that I like to do this by tracing the graph going left to right. And any time I need to switch directions, I draw narrow. Right? So if I am starting right here, I go down, right. It's that one going down and groups woman, There we go. Going down, down. I'm still going down dis faster at that point rate And then versus here If I'm going down Down, down, down, down, down, down, down, Down Right. I'm still kind of going down just less and less each time So if I'm looking at this, there's no where On this graph where I'm increasing right, This is a constantly decreasing function. And so then decreasing. I can pretty much say if you want to write an interval notation you can, um it's from negative. Infinity till zero. And then again from zero until infinity, right? Everything but zero. So if you want to write that, you can as well okay. And then this forest constant. We're looking for straight lines or single points. Um, and sense of secrecy everywhere. The only place we have to check is zero, but zero. It doesn't exist, because if we did 1/0, right, that's undefined. That's why we have the Assam totes there. Okay, um and if you wanted to check and make sure this was decreasing everywhere, plugging a couple points, right, if I plugged in like negative 10 I'd have won over negative 10 or negative one tense versus if we plugged in. Negative five. That's one over. Negative five. Excuse me. Um, that's native 1/5 rate. And, um so this is like a negative 0.1. This is point to negative point to raise. So it's continue decreasing, and then same over here. Right? If you plugged in one, you get 1/1, and then you get less and less. 1/2 1/3 1 4th 1/5 It's getting smaller and smaller. Smaller as it's going 20 Right. Um, anyway, so I hope this helps

Suppose we had a random function on the once to decide where dysfunction is increasing or decreasing. So Amanda MiGs on who f of X, equals four minus X squared. If we were cool, we know that full a graph to be increasing F X needs to be greater than zero. That's increasing. And if ffx is less than zero, go on is decreasing. So what we need to do you to find first derivative of a function so F that's X. The first vegetable four is just zero on the first inevitable minus. Don't to get outside minus X squared is minus two X So this means our f that X is just minus two X. No, but no, that we know if we have a critical value, we know that either side of this critical value glove will either be increasing or decreasing. Okay. And of course, we know that when f Dash X equals oh, we have a critical value. Okay, that's very important. So now if we equate or Dash's ex zero, we have minus two X equals zero, which means that X is just equal to Zillow, and this is how critical value. So we now need to look at both science of zero. So we know that even side of zero the glove is even going to be increasing or decreasing. So if you go back 12 who's here? We know that when f slash X is greater than zero is increasing on when f dash Texas lessons and it's decreasing. So if we look at our FBI sex, which is minus two X, we know if X is greater than zero, we know f slash x is going to be lessons. Oh, because we have negative two times a positive number, which will give you a negative number on. Similarly, we know that if X was less than zero I a negative number, you know that F of X would be greater because you're going toe to negative numbers. We have negative two times a negative number, which gives us a positive number. So this means when X is greater than zero effort Beck's lessons. Oh, which means if we look back here when Effort X is less than zero, it's going to be decreasing. Andi similarly here we know that when exes less than zero f of X is going to be greater than zero. So we know here that it's going to be increasing and these are about two ounces

So if we were given a function when he wanted to find out where in this function are the areas off increasing and decreasing intervals? How would we go about? Well, we know that when after X is greater than zero, the function is increasing. And when F dash rex is lessons up, it is decreasing. So notice we only have the function and not the first derivative. So that's clearly what we need to work out. So we need to get the first derivative. So the first rivet before it's just zero on the first derivative of negative X squared. Remember, the minus side is minus Teoh X, and this is, of course, just negative. Two X. Now we simply need to do is substitute for Data X into our increase in on decrease in inequalities. So let's stop with increase in verse, so f x these to be greater than zero for it to be increasing or f slash X is minus two x Once we divide by to on both sides, we get minus X is greater than zero. Now remember we two times by negative one. But once we times or divide by negative one, we always flip the inequality side. So once we times by negative one, we get X. But because we're times in by elective, we need to fit the inequality side. So I was actually going to be X is less than zero when it's increasing. But notice we need an interval. So we need to include infinity. So for values off X are left, Enzo is going to be increasing. So we need to include more negative infinity because negative infinity is less than zero. So our interval will be negative. Infinity and and so because it's not including zero, you don't use the square bracket. So that's the intergroup for increasing. Now you simply look at decreasing if the aspects is less than zero or aftershocks is negative. Two X lessons. We divide by two so we get negative X less than zero Now the active times by a negative one x And remember when we times by negative one we flip the inequality sign so I mean X is greater turns out so for bodies of X created and syrup F is decreasing, but remember we need interval. So for all values of X created and 0123 will be up to infinity. It's always going to be decreasing, so are bound will be between zero and infinity. So when X is between negative infinity and zero, it's increasing on. When it's between zero and infinity, it's decreasing.

We want to determine where our function is increasing and where it's decrease, so we need to find the critical points. So let's take the derivative of our function, which is to acts. Critical points would be any point where the derivative is zero or non differential. Since we started with a polynomial function, it's never non differential. We could have critical points where the derivative equals zero and dividing by two lets us know there is a critical point of zero. Let's check around that point to see what sign F prime is. If I pick a point less than zero like negative one. The derivative is negative. If I pick a value like one, the derivative is positive. So we've got this decreasing region here to the left, increasing to the right. So the function decreases from negative infinity to zero and increases from zero to infinity.


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