The function 𝑓 has the property that 𝑓 of 𝑥 and 𝑓 double prime of 𝑥 are positive, while 𝑓 prime of 𝑥 is negative for all 𝑥 is greater than zero. Which of the following could be the graph of 𝑓?
And then we’ve been given four graphs to choose from. Let’s go to the question and identify what it means for each of these values to be either positive or negative. Firstly, we’re told that 𝑓 of 𝑥 is positive. It’s greater than zero, the values of 𝑥 greater than zero. This means its range is 𝑓 of 𝑥 is greater than zero. And the graph must lie only in the first quadrant. Secondly, we’re told 𝑓 prime of 𝑥 is negative. It’s less than zero for all values of 𝑥 greater than zero.
If the derivative of a function is less than zero, it means that function is decreasing. Geometrically, it means the graph of the function must be sloping downwards. Finally, we’re told that the second derivative 𝑓 double prime of 𝑥 is positive. It’s greater than zero. Now, if 𝑓 double prime of 𝑥 is positive, the graph of the original function 𝑓 of 𝑥 is concave upwards. Another way of thinking about that is that the first derivative 𝑓 prime of 𝑥 is increasing. So which of our graphs satisfy all three criteria?
We said that the graph lies in the first quadrant. Well, A, B, C, and D satisfy these criteria. So this first one isn’t a lot of use. The second criteria tells us that the graph must be sloping downwards. While we can see graph A is sloping upwards, graph B is indeed sloping downwards, as is Graph C. Whereas graph D is, once again, sloping upwards. And so, we can disregard graph A and graph D. Our third piece of information told us that the graph is concave upwards. In other words, the first derivative is increasing. Let’s have a look of what that actually means.
The first derivative tells us the slope of the graph. So the slope of our graph must be increasing. Lets add tangents to the curve in graph B. When 𝑥 is equal to 0.2, we can see that the tangent to the graph is quite steep. However, it would have a negative value. So whilst it’s going to be a large value, it’s going to be a large negative value. Whereas when 𝑥 is equal to seven, the slope of the tangent seems much shallower. It’s still negative but now very close to zero. It’s getting less and less negative.
So actually, the slope 𝑓 prime of 𝑥 is indeed increasing for graph B. So it looks like graph B could be the one we’re interested in. Let’s double check graph C. If we add tangents to the curve for graph C, we see that the slope is getting more and more negative. 𝑓 prime of 𝑥 is decreasing. And so, 𝑓 double prime of 𝑥 is less than zero. The graph we’re interested in then and the one that satisfies all three criteria is graph B.