So you still have g prime of x is less than zero. G prime as less than zero so let me write that down. Still get a negative value so this is going to be negative. Value to the third power, well, you're going to To get negative values in this denominator, the denominator is still going to be, you take a negative We're still going to get negative values in this denominator right over here. So if x, we could just sayįor example, if x was one, one minus two is negative one. Going to be positive and so let's see, if we have x minus two where x is greater than With g prime of x here? Well, once again, x squared, anything greater than a zero and it says we're not including So this is the interval from zero to two, the open interval. Now, let's take the intervalīetween zero and two right over here. To know when it's decreasing, we would know it's definitelyĭecreasing over that interval. g prime of x is less than zero or if we cared or if we want So on this interval, on this interval, I'll write it like this. Positive divided by a negative so g prime is going to be negative so let me write that down. Well, a negative number to the third power is going to be a negative number so that right over there Get a negative number and then you take it to the third power. You subtract two from it, you're still going to Now, what about the denominator? You take a negative number, If you take any negative value squared, you're going to get a positive value so this is going to be positive. 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 Let's think about the interval between, between negative infinity and zero. So let's think about, let's first think about this interval. The critical values or on either side of the critical values. 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 xĮquals zero right over there and we have a critical point at x equals, at x equals two right over there. What g prime is doing in the intervals between Let's put them on a number line and let's just think about Points or critical values here and what I'm going to do 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 Place where g prime of x is equal to zero and where is g prime of x undefined? Well, it's going to be undefined if the denominator becomes undefined. Prime of x equal to zero is getting the numerator equal to zero and that's only going to happen if x squared is equal to zero So when is g prime of x equal to zero? Well, the way to get g Where the sign could change, the sign of g prime could change. Is those are the places, those are possible places Points or critical values and why those are relevant 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 So critical, critical points for g and just to remind ourselves Look at the critical points or the critical values for g. To be greater than zero or we could do it a littleīit more methodically. You might just want to inspect kind of the structure of thisĮxpression and think about, well, when is that going 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 Intervals is g increasing, that's equivalent to saying, on which intervals is the firstĭerivative with respect to x on which intervals is that How do we figure out when g is increasing? Well, the answer is all we need is g prime which they do give us. On which intervals is g increasing? Well, at first you might say, Also let g prime, the derivative of g, be defined as g prime of x is equal to x squared over x Let g be a function defined for all real numbers.
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