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 zero.