The graph of 𝑓 prime if shown. Which of the following is a possible graph of 𝑓? Is it a), b), c), or d)?
Note that the graph of 𝑓 prime that is given to us in the question is a curve known as a parabola. This parabola has its axis of symmetry parallel to and, in fact, equal to the 𝑦-axis. Therefore, it represents a quadratic equation of the form 𝑘 two 𝑥 squared plus 𝑘 one 𝑥 plus 𝑘 zero. Where 𝑘 two, 𝑘 one, and 𝑘 zero are constants, with 𝑘 two not equal to zero.
So we have found that 𝑓 prime is a quadratic function. In order to find the function 𝑓 of 𝑥 from the function 𝑓 prime of 𝑥, we will find the indefinite integral of the function 𝑓 prime of 𝑥 with respect to 𝑥. As the indefinite integral of the function 𝑓 prime of 𝑥 with respect to 𝑥 equals 𝑓 of 𝑥 plus 𝑐, where 𝑐 is the constant of integration. To do this, we will apply the power rule for integration to each term in the equation for 𝑓 prime of 𝑥 that we have formulated.
Doing so, we obtain that 𝑓 of 𝑥 equals 𝑘 two 𝑥 cubed over three plus 𝑘 one 𝑥 squared over two plus 𝑘 zero 𝑥 plus 𝑐, where 𝑐 is a constant. The highest power of 𝑥 in 𝑓 of 𝑥 is three. With coefficient equal to 𝑘 two over three. Note that 𝑘 two is not equal to zero implies that 𝑘 two over three is not equal to zero. Hence, 𝑓 of 𝑥 is a cubic function.
Recall that a general cubic function is either of the form one or two. It is of the form one if the coefficient of 𝑥 cubed is positive. And it is of the form two if the coefficient of 𝑥 cubed is negative. We have that 𝑘 two, the coefficient of 𝑥 squared in 𝑓 prime of 𝑥, is positive. Since 𝑓 prime of 𝑥 is a parabola with an upward opening. Hence, 𝑘 two over three, the coefficient of 𝑥 cubed in 𝑓 of 𝑥, is also positive. Therefore, the graph of 𝑓 of 𝑥 is a cubic curve of the form illustrated as number one.
From this, we can deduce that the correct answer to the question is not option c) or option d). As the curves in those options do not resemble the shape of the cubic curve. The graph in option c) represents a quadratic curve with a negative 𝑥 squared coefficient. And the graph in option d) represents a quartic curve, where the highest power of 𝑥 is four.
Option a) illustrates a graph with a cubic curve, which has a minimum point on the left and a maximum point on the right. This resembles the form of the cubic curve illustrated as number two. Therefore, option a) is not the correct answer.
Option b) illustrates a cubic curve with a maximum point on the left and a minimum point on the right. This indeed resembles one of the cubic curves illustrated as number one. Hence, option b) is a possible graph of 𝑓. Before we finish, let’s quickly have a look at some of the other properties of the graph of the function 𝑓 prime of 𝑥. And see if they can be used to arrive at the correct answer or at least rule out any incorrect options.
Note that the graph of the function 𝑓 prime of 𝑥 also tells us that 𝑓 prime of 𝑎 equals zero and 𝑓 prime of 𝑏 equals zero. Therefore, the slope of the function 𝑓 of 𝑥 must be zero at the points 𝑥 equals 𝑎 and 𝑥 equals 𝑏. We can see that the slope of 𝑓 of 𝑥 is zero at 𝑥 equals 𝑎 and 𝑥 equals 𝑏 in all options, apart from option c).
In option c), the slope of 𝑓 at 𝑥 equals 𝑎 is positive and at 𝑥 equals 𝑏 is negative. Therefore, option c) is not the correct answer according to this method. Moreover, notice that the graph of 𝑓 prime of 𝑥 tells us that 𝑓 prime of 𝑥 is less than zero for values of 𝑥 between 𝑎 and 𝑏. Therefore, the slope of 𝑓 of 𝑥 is negative for values of 𝑥 between 𝑎 and 𝑏. The only option that correctly portrays this is option b). We can clearly see that the slope of 𝑓 is negative for values of 𝑥 between 𝑎 and 𝑏 in option b). And that this is not the case in any of the other options.
Lastly, note that 𝑓 prime of 𝑥 is greater than zero if 𝑥 is less than 𝑎 or 𝑥 is greater than 𝑏. Therefore, the slope of 𝑓 is positive for all values of 𝑥 outside the interval from 𝑎 to 𝑏. Again, the only option that correctly portrays this is option b). So we have seen a few different methods which we can use to arrive at the correct answer, which is option b).