Increasing, decreasing, positive or negative intervals (video) | Khan Academy

Video transcript

– [ Voiceover ] What I hope to do in this video recording is look at this graph yttrium is equal to f of x and think about the intervals where this graph is positive or negative and then think about the intervals when this graph is increasing or decreasing. So beginning let ‘s just think about when is this serve, when is this function positive ? Well cocksure means that the value of the officiate is greater than zero. It means that the value of the function this means that the function is sitting above the x-axis. So it ‘s sitting above the x-axis in this plaza right over hera that I am highlighting in scandalmongering and it is besides sitting above the x-axis over here. And if we wanted to, if we wanted to write those intervals mathematically. Well let ‘s determine, let ‘s say that this point, let ‘s say that this orient right over here is ten equals a. Let ‘s say that this right over hera is ten equals bel and this right over hera is x equals c. then it ‘s positive, it’s positive equally hanker as adam is between a and bacillus. At ten equals a or at x equals b the rate of our affair is zero but it ‘s positive when ten is between a and boron, a and bacillus or if x is greater than c. ten is, we could write it there, hundred is less than ten or we could write that ten is greater than c. These are the intervals when our function is positive. Let me write this, f of x, fluorine of x positive when adam is in this interval or this interval or that time interval. so when is f of x negative ? Let me do this in another tinge. F of x is going to be negative. well, it ‘s gon sodium be negative if x is less than a. So this is if x is less than a or if x is between b and deoxycytidine monophosphate then we see that degree fahrenheit of ten is below the x-axis. F of x is down hera thus this is where it ‘s negative. so here or, or ten is between b-complex vitamin or degree centigrade, x is between boron and c. And I ‘m not saying less than or equal to because at b or c the measure of the function f of boron is zero, fluorine of coulomb is zero. That ‘s where we are actually intersecting the x-axis. So that was reasonably aboveboard. nowadays let ‘s ask ourselves a different question. When is the function increasing or decreasing ? so when is f of ten, fluorine of adam increasing ? Well increasing, one way to think about it is every time that x is increasing then y should be increasing or another way to think about it, you have a, you have a positivist rate of switch of y with respect to x. We could tied think about it as think if you had a tangent note at any of these points. If you had a tangent occupation at any of these points the gradient of that tangent pipeline is going to be positive. But the easiest direction for me to think about it is as you increase adam you’re going to be increasing y. so where is the routine increasing ? Well I ‘m doing it in blue sky. So it ‘s increasing correct until we get to this point right over here, correct until we get to that point over there then it starts decreasing until we get to this point right over here and then it starts increasing again. It starts, it starts increasing again. So let me make some more labels here. So let ‘s say that this, this is x equals five hundred and that this justly over here, actually let me do that in green color, so let ‘s say this is ten equals d. immediately it ‘s not a, d, boron but you get the picture and lashkar-e-taiba ‘s say that this is x is equal to, x is peer to, let me redo it a fiddling moment, x is equal to e. X is adequate to e. therefore when is this function increasing ? Well it ‘s increasing if x is less than five hundred, x is less than five hundred and I ‘m not gon na say less than or peer to ’cause right at adam equals d it looks like fair for that moment the slope of the tangent note looks like it would be, it would be changeless. We ‘re going from increasing to decreasing indeed right at d we ‘re neither increasing or decreasing. But then we ‘re besides increasing, so if x is less than d or adam is greater than e, or adam is greater than e. And where is fluorine of adam decreasing ? So fluorine of ten, let me do this in a different color. When is, let me pick a mauve, so f of adam decreasing, decreasing well it ‘s going to be correctly over here. It ‘s gon sodium be correctly between five hundred and e. Between x equals five hundred and ten equals e but not precisely at those points ’cause at both of those points you ‘re neither increasing nor decreasing but you see good over here as ten increases, as you increase your x what ‘s happening to your yttrium ? If you go from this point and you increase your x what happened to your y ? Your yttrium has decreased. You increase your x, your y has decreased, you increase your ten, y has decreased, increase x, y has decreased all the way until this point over here. So f of adam is decreasing for x between five hundred and e. So hopefully that gives you a sense of things. Notice, these are n’t the same intervals. That we are, the intervals where we ‘re convinced or negative do n’t absolutely coincide with when we are increasing or decreasing. So it ‘s very significant to think about these individually flush though they kinda sound the same.

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