The graph of 𝑓 is shown in the figure and 𝑓 is twice differentiable. Which of the following has the largest value? (I) 𝑓 of zero, (II) 𝑓 prime of zero, (III) 𝑓 double prime of zero. And the options for our answer are (a) (II) only, (b) (I) only, (c) (III) only, and (d) (I) and (III).
Here, we have the point 𝑓 of zero. We also need to consider 𝑓 prime of zero and 𝑓 double prime of zero, in other words, the first derivative of 𝑓 at this point and the second derivative of 𝑓 at this point. So, firstly, what observations can we make about 𝑓 of zero? Well, clearly, it’s less than zero; it’s negative. And that’s all we can really say about this point. So let’s think about 𝑓 prime of zero, the first derivative of 𝑓 at this point.
Remember that the first derivative gives us the slope. So we’re looking at the slope at this point. At this point, the slope looks something like this. So we remember at this point that a positive slope is a slope that goes from the bottom left to the top right. And a negative slope is a slope that goes from the top left to the bottom right. We can then observe, at the point 𝑓 of zero, we see a positive slope. And because the slope at this point gives us the first derivative at this point, we can say that 𝑓 prime of zero must be positive.
So we can already see that 𝑓 prime of zero is going to be bigger than 𝑓 of zero. But we still need to consider 𝑓 double prime of zero, the second derivative of 𝑓 at 𝑥 equals zero. If we draw more tangent lines around the point that we’re interested in, we can see that, around this point, the slope of the tangent lines is actually decreasing. And when this happens, we say that 𝑓 is concave down. And because the slope of these tangent lines is decreasing, what we’re saying is that the first derivative is decreasing. And because the second derivative is just the derivative of the first derivative, the slope of the first derivative tells us about the second derivative.
So where the first derivative is decreasing, its slope is negative. So when 𝑓 is concave down, the first derivative is decreasing. And so, the second derivative is negative. So from the graph alone, we’ve been able to tell that, at 𝑥 equals zero, 𝑓 is negative, 𝑓 prime is positive, and 𝑓 double prime is negative. So even though we don’t know what these values are, 𝑓 prime of zero is the only positive value. So 𝑓 prime of zero must be the largest. So the answer is (II) only.