Intervals on which a function is increasing or decreasing
Finding decreasing interval given the function
- [Instructor] Let's say we have the function F of X is equal to X to the sixth, minus three X to the fifth. And my question to you is using only what we know about derivatives. Try to figure out over what interval or intervals is this function decreasing? Pause the video and try to figure that out. All right, now let's do this together. So we know that a function is decreasing when its derivative is negative. Or another way to say it, It's going to be decreasing when F prime of X is less than zero. So what is F prime of X? Well, we could use the derivative rules and drew the properties we know. We apply the power rule to X, to the sixth, We bring the six out front or multiply the one coefficient here, times six, to get six, X to the fifth detriment that exponent, minus bring the five times the three minus 15, X to the, we detriment the five, so X to the fourth. And we need to figure out over what intervals is this going to be less than zero. And now let's see how we can simplify this a little bit. Both of these terms are divisible by X to the fourth, and they're both divisible by three. So let's factor out a three X to the fourth times, you factor out of three X to the fourth year, you're left with a two X, and then over here you have minus five has to be less than zero. Any interval where this is true, we are going to be decreasing. Now, how do we get this to be less than zero? Well, if I take the product of two things and it's less than zero, that means that they have to have different signs, either one's positive and the other's negative or one's negative and the other's positive. So we have two situations. So we could say either, either, three X to the fourth is greater than zero and, and two X minus five is less than zero. So that's one situation. I'll do some in a different color, or I'll do this one in a different color, three X to the fourth is less than zero and two X minus five is greater than zero. Actually let me stay on the second case first. Are there any situations where three X to the fourth can be less than zero? You take any number, you take it to the fourth power even if it's a negative, it's going to become a positive. So you can't get a negative expression right over here. So actually the second condition is impossible to obtain. You can't get any situation for any X where three X to the fourth is less than zero. So we can rule this one out. And so this is our best hope. So under what conditions is three X to the fourth, greater than zero. Well, if you divide both sides by three, you get X to the fourth is greater than zero. And if you think about it, this is gonna be true for any X value that is not equal to zero. Even if you have a negative value there, if you have a negative one, you take it to the fourth power becomes a positive one. Only zero will be equal to zero when you take it to the fourth power. So one way we could say this is going to be true for any non zero X, or we could just say, X does not equal zero, and, this is a little bit more straightforward, we add five to both sides we get two X is less than five, divide both sides by two, you get X is less than five halves. So it might be tempting to say, all right, the intervals that matter are all the exes less than five halves, but X cannot be equal to zero. Now, is that the entire interval where our function is decreasing? Well, let's think about what happens at zero itself. We're decreasing over the interval from negative infinity all the way up to zero. And we're also decreasing from zero to five halves. And so for decreasing right to the left of zero, we're decreasing right to the right of zero, we're actually going to be decreasing at zero at we're also gonna be decreasing at zero as well. So there's something interesting here, even though the derivative at X equals zero, is going to be equal to zero, we are still decreasing. And so the interval that we care about the interval over which we're decreasing is just X is less than five halves. And we can see that by graphing the function, I graphed it on Desmos, and you can see here that the function is decreasing from negative infinity, it's decreasing at a slower and slower rate we get to zero still decreasing to the left of zero, and then it continues to decrease to the right of zero. So any, any value, any X value to the left of zero the value of the function is going to be larger than F of zero, and X to the right of zero. The value of the function is going to be less than the function at zero. So it's actually decreasing through zero, even though the slope of the tangent line at zero is zero, even though it's non negative and then we keep decreasing so we're decreasing for all values of X, less than five halves, which you can see visually here.