Integrate Initial Justification with airSlate SignNow

Integrate initial justification

- [Instructor] "The differentiable function f "and its derivative f prime are graphed." So, let's see. We see the graph of y is equal to f of x here in the blue. And then f prime we see in this brownish-orangish color right over here. "What is an appropriate calculus-based justification "for the fact that f is decreasing "when x is greater than three?" So, we can see that that actually is indeed the case. When x is greater than three, we see that our function is indeed decreasing. As x increases, the y value, the value of our function, decreases. So, a calculus-based justification, without even looking at the choices, well, I could look at the derivative. And we're going to be decreasing if the slope of the tangent line is negative, which means that the derivative is negative. And we can see that for x is greater than three, the derivative is less than zero. So, my justification, I haven't even looked at these choices yet. I would say for x is greater than three, f prime of x is less than a zero. That would be my justification, not even looking at these choices. Now let's look at the choices. "f prime is decreasing when x is greater than three." Now, this isn't right. What we care about is whether f prime is positive or negative. If f prime is negative, if it's less than zero, than the function itself is decreasing. The slope of the tangent line will be negative. f prime could be positive while decreasing. For example, f prime could be doing something like this. And even though f prime would be decreasing in this situation, the actual value of the derivative would be positive, which means the function would be increasing in that scenario, so I would rule this one out. "For values of x larger than three, as x values increase, "the values of f of x decrease." Now, that is actually true. This is actually the definition that f is decreasing. As x values increase, the values of f of x decrease. But this is not a calculus-based justification, so I am going to rule this one out as well. "f prime is negative when x is greater than three." Well, that's exactly what I wrote up here. If f prime is negative, then that means that our slope of the tangent line of our original function f is going to be downward sloping, or that our function is decreasing, so this one is looking good. And this one right over here says, "f prime of zero is equal to negative three," so they're just pointing out this point. This isn't relative to the interval that we care about, or this isn't even relevant when x is greater than three, so we definitely wanna rule that one out. Let's do one more of these. So, here we're told, "The differentiable function g "and its derivative g prime are graphed." So, once...