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# Finding increasing interval given the derivative

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.1 (EK)

## Video transcript

let G be a function defined for all real numbers also let G prime the derivative of G be defined as G prime of X is equal to x squared over X minus 2 to the third power on which intervals is G increasing well at first you might say well they don't even give us G how do we figure out when G is increasing well the the answer is all we need is G prime which they do give us and saying on which intervals is G increasing that's equivalent to saying on which intervals is the first derivative with respect to X on which intervals is that going to be greater than zero if your rate of change with respect to X is greater than zero if it's positive then your function itself is going to be increasing and so there's a couple of ways that we could approach this you might just want to inspect the kind of the structure of this expression and think about well when is that going to be greater than zero or we could do it a little bit more methodically we could say well let's look at the critical points or the critical values for G so critical critical points for G and just to remind ourselves what critical points are that is when G prime of X is equal to zero or G prime of X is undefined is undefined and we have videos on critical points or critical values and why those are relevant is those are the places those are the possible places where the sign could change the sign of G prime could change so when is G prime of X equal to zero well the way to get G prime of x equal to zero is getting the numerator equal to zero and that's what I'm going to happen if x squared is equal to zero or if X is equal to zero so that's the only place where G prime of X is equal to zero and where is G prime of X undefined what's going to be undefined if the denominator becomes undefined the denominator becomes undefined if the denominator is zero and so that's going to happen if X minus two is equal to zero X minus two is equal to zero or X is equal to two so we have two critical points or critical values here and what I'm going to do is I'm going to graph them let's put them on a number line and let's just think about what G Prime is doing in the intervals between the critical points so let's start at zero one two three and then let's go to negative one and we have a critical point at let me do that in magenta we have a critical point at x equals zero right over there and we have a critical point at x equals at x equals two right over there and so let's think about what G prime is doing in these in the intervals between the critical values or on either side of the critical values so let's think about let's first think about this interval let me do it in this purple color let's think about the interval between between negative infinity and zero so if we think about this interval so negative infinity and zero that open interval well if we look at G prime the numerator is still going to be positive you take any negative value square it you're gonna get a positive value so this is going to be positive now what about the denominator you take a negative number you subtract two from it you're still going to get a negative number and then you take it to the third power well the negative number to the third power is going to be a negative number so that right over there is going to be negative so you're gonna have a positive divided by a negative so G prime is going to be negative so let me write that down so on this interval on this interval alright like this G prime of X is less than zero or if we cared if we want to know what it's decreasing we would know it's definitely decreasing over that interval now let's take let's take the interval between zero and two all right over here so this is the interval from zero to two the open interval so what's going to go on with G prime of X here well once again x squared anything greater than a zero and it says we're not including zero in this interval well this is for sure are going to be positive and so let's see if we have X minus 2 where X is greater than zero but less than two so if X we could just say for expos 1 1 minus 2 is negative 1 we're still going to get negative values in this denominator right over here so since we're still going to get negative values in this denominator the denominator is still going to be take a negative value to the third power well you're going to still get a negative value so this is going to be negative so you're still going to have G prime as less than zero so let me write that down so you still have G prime of X is less than zero and then let's take the interval above let's take the interval from 2 to infinity 2 to infinity well the numerator is positive it's always going to be positive 4 for any X not being equal to 0 and this denominator you're taking values greater than 2 subtracting 2 from it which is still going to give you a positive value you take the 3rd power it's all going to be positive it is all going to be positive so this is the interval where G prime of X is greater than 0 so on which intervals is G increasing well that's where G prime of X is greater than 0 so it's going to be from 2 from 2 to infinity or we could just write it like this we could write x is greater than 2 either way if for either of these G prime of X is greater than 0 and your function G is going to be increasing