Video Transcript
True or False: When drawing the function 𝑓 of 𝑥, if the second derivative of 𝑓 of 𝑥 is positive at 𝑥, then the function 𝑓 of 𝑥 is concave up at 𝑥.
In this example, we’re asked to consider the second derivative. When sketching the graph of a function, the first derivative and second derivative can help us determine a lot of information about our curve. The second derivative in particular will help us determine the intervals for which our function is concave upward or concave downward, whereas the first derivative helps us determine on what interval the function increases or decreases.
We will now recall the properties of the second derivative. First, the function 𝑦 equals 𝑓 of 𝑥 must be twice differentiable on the open interval from 𝑎 to 𝑏. Then, if the second derivative of 𝑓 of 𝑥 is positive for any 𝑥 on the open interval from 𝑎 to 𝑏, then 𝑓 is concave upward on that open interval. If the second derivative of 𝑓 of 𝑥 is negative for any 𝑥 on the open interval from 𝑎 to 𝑏, then 𝑓 is concave downward.
It is worth noting that if the function 𝑓 of 𝑥 has a critical point when 𝑥 equals 𝑐, then we can use the second derivative test to attempt to categorize this point. If the second derivative at 𝑐 is positive, the point is a relative minimum. If the second derivative at 𝑐 is negative, the point is a relative maximum. However, if the second derivative at 𝑐 equals zero, the point can be a relative minimum, a relative maximum, or a point of inflection. In this case, we would need to check the first derivative on either side of 𝑐 to determine its nature.
In this example, we are given a statement that claims that if the second derivative of 𝑓 of 𝑥 is positive at 𝑥, then the function is concave up at 𝑥. According to the first property of second derivatives that we have recalled, this is a true statement.
