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In an episode of The Simpsons television show, Homer reads from a newspaper and announces "Here's good news! According to this eye-catching article, SAT scores are declining at a slower rate." Interpret Homer's statement in terms of a function and its first and second derivatives.

$f^{\prime}(x) < 0$ and $f^{\prime \prime}(x) > 0$

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Missouri State University

Oregon State University

Harvey Mudd College

University of Nottingham

So we're told in this question that s A T scores are decreasing at a slower rate. So what that means is that we're decreasing but we're actually not as decreasing as much as we were before. So if we were to look at a graph that's decreasing at a slower rate at some point would be decreasing, decreasing decreasing and then we're decreasing less and less and less like this or going towards a positive slope, but we're still decreasing. So this is what a graph would look like, where we could be saying, somewhere over here we're decreasing less and less and less, and then we start increasing. So what this tells us about our first derivative is that we know that our first derivative f prime of X is going to be negative since we're decreasing whenever we're decreasing, our first derivative is going to be negative. So we know that it's going to be less than zero. Our second derivative is actually going to be positive. And the reason for this is because we're concave up at this point where we're decreasing less and less. If we were decreasing more and more, we would be concave down. We can also think of this as if we were looking at a graph of our first derivative, we're decreasing less. So we start at some negative value and we're getting closer and closer to zero, which means that we have a positive slope for our first derivative, which means that the derivative of that of our first derivative has to be positive at that point. So those are two different ways to know that we're going to be concave up at that point in our second derivative is going to be positive.