Justification using first derivative

- [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 again, g is in this bluish color, and then g prime, its derivative, is in this orange color. "What is an appropriate calculus-based justification "for the fact that g has a relative minimum point "at x is equal to negative three?" And we could see here, when x is equal to negative three, it looks like g is equal to negative six, and it looks like a relative minimum point there. So, what's the best justification? So, once again, without even looking at the choices, I would say a good justification is before we get to x equals negative three, before we get to x equals negative three, our derivative, and this is a calculus-based justification, before we go to x equals negative three, our derivative is negative. And after x equals negative three, our derivative is positive. That would be my justification. Because if our derivative is negative before that value, that means that we are downward sloping before that value. And if it's positive after that value, that means we're upward sloping after that, which is a good justification that we are at relative minimum point right over there. So, let's see, "The point where x equals negative three "is the lowest point on the graph of g "in its surrounding interval." That is true, but that's not a calculus-based justification. You wouldn't even have to look at the derivative to make that statement, so let's rule that one out. "g prime has a relative maximum at zero comma three." At zero comma three, it actually does not. Oh, g prime, yes, g prime actually does have a relative maximum at zero comma three, but that doesn't tell us anything about whether we're at a relative minimum point at x equals negative three, so I would rule that out. "g prime of negative three is equal to zero." So, g prime of negative three is equal to zero, so that tells us that the slope of the tangent line of our function is going to be zero right over there, but that by itself is not enough to say that we are at relative minimum point. For example, I could be at a point that does something like this where the slope of the tangent line is zero, and then it keeps increasing again, or it does something like this and it keeps decreasing. So, even though you're at a point where the slope of your tangent line is zero, it doesn't mean you're at a relative minimum point, so I would rule that out. "g prime crosses the x-axis from below it to above it "at x equals negative three." g prime crosses the x-axis from below it to above it. Yep, and that's the argument that I made, that we're going from below the x-axis, so g prime goes from being negative to positive, which means the slope of the tangent lines of our points as we approach x equals negative three go from being downward sloping to upward sloping, which is an indication that we are at a relative minimum point.