Which of the following curves is convex downwards for the set of all 𝑥s that are real numbers?
We say that a curve is convex downward when the slope is always increasing. We need to determine along which of these curves is the slope always increasing. We can use derivatives to help us do that. By taking the first and second derivative, we can tell if the slope is increasing or decreasing.
For function a, if we take the first derivative, the derivative of three is zero, and the derivative of negative 𝑥 squared is negative two 𝑥. At this point, we can already see that the slope is not going to be increasing. When the second derivative is negative two, that means for all 𝑥-values the slope is decreasing.
If we take the first derivative of function b, the derivative of three is zero, and the derivative of negative 𝑥 cubed is negative three 𝑥 squared. And we’ll take the second derivative of negative three 𝑥 squared, which is negative six 𝑥. Now negative six 𝑥 would sometimes yield a positive value. When 𝑥 is negative, when 𝑥 is less than zero, function b is convex downward. Its slope is increasing when 𝑥 is negative. And when 𝑥 is positive, its slope is decreasing. This function is partially convex downward. It’s convex downward when 𝑥 is less than zero. However, our job is to find a convex downward curve for all 𝑥-values. So, this is not the solution we’re looking for.
Function c, we need to take its first derivative. Its derivative is going to be negative four 𝑥 cubed. To check for an increasing slope, we need the second derivative, which is negative 12𝑥 squared. And here, no matter what we plug in for 𝑥, it’s going to yield a positive result. And then, we have to multiply that by a negative. And a positive times a negative will always give us a negative. Function c is decreasing at every point. It’s actually convex upward at all points.
Let’s consider function d. Its first derivative is four 𝑥 cubed, which makes its second derivative 12𝑥 squared. 12𝑥 squared always returns a positive result. And that means that at every point, it’s slope is increasing. And it is a convex downward function. 𝑓 of 𝑥 equals three plus 𝑥 to the fourth power is convex downward.