Video Transcript
In the figure shown, for which of the four given points on the graph is d𝑦 by d𝑥 less than zero and d squared 𝑦 by d𝑥 squared less than zero?
So we have points 𝐴, 𝐵, 𝐶, and 𝐷 on this graph. And for this question, we need to think about what we should be looking for on this graph that indicates when the first derivative and second derivative are both less than zero. We’ve not been given the function. We’ve only been given the graph. So let’s think about the graphs of the first and second derivatives for a generic function 𝑓 of 𝑥.
If 𝑓 of 𝑥 looks like this, let’s have a think about what the graph of the derivative would look like, remembering that the first derivative is the slope of 𝑓 of 𝑥. And so where 𝑓 of 𝑥 has a positive slope, the graph of the first derivative is positive. And where 𝑓 of 𝑥 has a negative slope, the graph of the first derivative is negative. So we can see that d𝑦 by d𝑥 is less than zero where 𝑓 of 𝑥 has a negative slope.
So now we think about the second derivative. One way we can think of the second derivative is as the slope function of the first derivative. So where the slope of the first derivative is negative, the graph of the second derivative is negative. And similarly, where the slope of the first derivative is positive, the graph of the second derivative is positive.
If we compare the graph of the second derivative with 𝑓 of 𝑥, we can see that the second derivative is negative where 𝑓 of 𝑥 is concave down. This is because the slope of 𝑓 of 𝑥 is decreasing over that interval, meaning that the first derivative is decreasing. So the second derivative is negative.
Okay, so in our question, we’re looking for where the first and second derivatives are less than zero. We’ve seen that the first derivative is less than zero, where 𝑓 of 𝑥 has a negative gradient. In other words, 𝑓 of 𝑥 is decreasing. And we’re also looking for where the second derivative is less than zero, so where 𝑓 of 𝑥 is concave down.
So on our graph, which of these points is both decreasing and concave down? Well, this part of the graph is concave down and this part is decreasing. And so the point where the graph is both concave down and decreasing is point 𝐵.
