The graph of the function 𝑓 is shown in the figure. For which of the following values of 𝑥 is 𝑓 prime of 𝑥 negative and decreasing?
Let’s consider each of these properties of the first derivative 𝑓 prime of 𝑥 in turn. For the first derivative 𝑓 prime of 𝑥 to be negative at a point, this means that the slope of the tangent to the curve at that point must be negative. We can draw in approximate tangents to the curve at each of the points 𝑎, 𝑏, 𝑐, and 𝑑 and determine the sign of their slope. We see, for example, that the slope of the tangent to the curve at point 𝑎 is positive because the tangent is sloping upwards. So the first derivative 𝑓 prime of 𝑥 will be positive not negative at point 𝑎.
However, at point 𝑏, if we sketch in a tangent to the curve here, we can see that it does slope downwards. And therefore, the first derivative 𝑓 prime of 𝑥 will be negative at point 𝑏. In the same way, if we sketch in the tangents at points 𝑐 and 𝑑, we see that the slope is negative at point 𝑐 but is positive at point 𝑑. And so we’ve eliminated both 𝑎 and 𝑑 from the possibilities.
The second requirement is that our slope 𝑓 prime of 𝑥 must be decreasing at a point. And this has all to do with the concavity of the curve. If the tangents to a curve lie above the curve itself in a particular region, then the curve is said to be concave downwards in that region. And its slope 𝑓 prime of 𝑥 is decreasing. In the left-hand sketch, we see that the slopes are positive. But the slopes are becoming less steep. And therefore, the slope is decreasing. Whereas in the sketch on the right, the slopes are negative and becoming more steep. But if they are becoming more steep, more negative, then the value is decreasing. Conversely, if the tangents to a curve lie below the curve itself in a particular region, then the shape of the curve is said to be concave upwards. And its slope 𝑓 prime of 𝑥 is increasing.
So we’re looking for a point where the slope of the tangent is not only negative but the tangent itself lies above the curve. Looking back at the original figure, we see that, at point 𝑏, the tangent we sketched is above the curve. And therefore the shape of the curve is concave downwards at this point. Whereas at point 𝑐, the tangent is below the curve. And so the curve is concave upwards at point 𝑐.
We conclude then that, of the four values of 𝑥, the only one at which the slope 𝑓 prime of 𝑥 is both negative and decreasing is 𝑏.