Video: Using the Graph of a Function to Determine the Sign of Its First and Second Derivatives

The function 𝑔 of π‘₯ is defined on the open interval (βˆ’3, 3) such that for all π‘₯, βˆ’3 < π‘₯ < 3, 𝑔′(π‘₯) > 0, and 𝑔″(π‘₯) > 0. Which of the following could be the graph of 𝑔(π‘₯) on (βˆ’3, 3)?

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Video Transcript

The function 𝑔 of π‘₯ is defined on the open interval negative three to three such that for all π‘₯ between negative three and three, 𝑔 prime of π‘₯ is greater than zero, and 𝑔 double prime of π‘₯ is greater than zero. Which of the following could be the graph of 𝑔 of π‘₯ on the open interval from negative three to three?

So, we’ve been given four graphs each of which might be the graph of a function 𝑔 of π‘₯. And we’re asked to use these graphs to determine properties about the first and second derivatives of this function. Let’s use a simple table to help us organise our thoughts.

Now, the first property we’re interested in is that 𝑔 prime of π‘₯, the first derivative of our function, must be greater than zero on the open interval from negative three to three. Now, remember, the first derivative of a function gives its slope. So, if we’re looking for 𝑔 prime of π‘₯ to be greater than zero throughout this interval, then we’re looking for a graph whose slope is always positive.

Remember that the slope of a curve is the same as the slope of the tangent to the curve at that point. So, by sketching in tangents to each graph as I’ve done on graph 𝐴, we can more easily see the slope of each curve. In the case of graph 𝐴, we see that each of the tangents are sloping downwards from left to right. And therefore, the slope of graph 𝐴 is always negative, meaning 𝑔 prime of π‘₯ will be less than zero on the open interval from negative three to three.

On graph 𝐡, however, we see that the slope of each of the tangents is positive; they’re sloping upwards from left to right. And therefore, 𝑔 prime of π‘₯, the first derivative, will be greater than zero for graph 𝐡.

On graph 𝐢, the sign of the slope of the tangents changes. It’s initially positive on the open interval from negative three to zero. But then, the slope of the tangents is zero at the turning point of the graph. That’s when π‘₯ is equal to zero. And then, the slope is negative on the open interval from zero to three. So, we can say that the sign of 𝑔 prime of π‘₯ on the open interval negative three to three for graph 𝐢. It isn’t consistently either positive or negative.

On graph 𝐷, finally, we see that the tangents are again sloping upwards. So, the first derivative 𝑔 prime of π‘₯ will be positive throughout the open interval negative three to three for graph 𝐷. Of the four graphs then, we’re left with two for which the first derivative 𝑔 prime of π‘₯ is always positive on this open interval, graph 𝐡 and graph 𝐷.

Now, let’s consider the second condition, which is that the second derivative 𝑔 double prime of π‘₯ must also be positive throughout this open interval. This sign of the second derivative of a function is related to the concavity of its graph. If 𝑔 double prime of π‘₯ is positive, then this means that 𝑔 prime of π‘₯ is increasing because the second derivative is the derivative of the first derivative. And if the derivative of a function is positive, then the function itself is increasing. So, if the derivative of the first derivative is positive, the first derivative is increasing.

We see that this is true for the curve sketched here. The slope of the tangents is changing from negative to zero to positive. So, the value of this slope will be getting larger, which means that 𝑔 prime of π‘₯ will be increasing. This type of concavity is referred to as concave up. And we further see that when a graph is concave up, each of the tangents to its graph lie below the graph itself.

The reverse of this is also true. When the second derivative of a function is negative, then its first derivative will be decreasing. We describe the shape of the curve in this region as concave down. And the tangents to the curve will lie above the graph itself. So, to determine whether each of these graphs are concave up or concave down, we just need to consider the position of their tangents relative to the position of the graph itself.

We can use the tangents we’ve already drawn. In graph 𝐴, we see that the tangents lie below the curve, meaning that this curve is indeed concave up. And therefore, its second derivative is positive. In graph 𝐡, the tangents again lie below the curve, so the same is true here. In graphs 𝐢 and 𝐷, however, the tangents lie above the curve itself throughout the open interval from negative three to three, which means that the functions in graphs 𝐢 and 𝐷 are concave down. And therefore, the second derivative 𝑔 double prime of π‘₯ will be negative for each of these.

We see then that the second derivative 𝑔 double prime of π‘₯ is positive for two of the given graphs, graph 𝐴 and graph 𝐡. Only graph 𝐡 then fulfills both of the required conditions. By considering the slope of its tangents, we saw that for graph 𝐡, 𝑔 prime of π‘₯ is always positive on the open interval from negative three to three. And then, by considering the concavity of the curve by looking at the position of its tangents, we saw that 𝑔 double prime of π‘₯ is also positive throughout the open interval negative three to three for graph 𝐡.

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