Determining Relative Extrema Using Differential Equations

Let $y = f(x)$ be a particular solution to the differential equation with the initial condition $f(0) = 1$. Does $f$ have a relative minimum, a relative maximum or neither at $x = 0$? Justify your answer.

In this problem, we are investigating the behavior of a function at a specific point, namely whether it has a relative minimum, maximum, or neither at $x = 0$. Such problems are commonly approached by examining the first and second derivatives of the function. The first derivative test can reveal if the function's slope changes from positive to negative or vice versa, indicating a potential extrema. Meanwhile, the second derivative test can confirm if that point is a minimum (concave up) or a maximum (concave down) by evaluating the concavity at that point.

In the context of differential equations, we often begin with a given differential equation and a particular initial condition, which can help determine specific solutions. This problem exemplifies these key concepts by asking whether there is a relative extremum at $x = 0$ based on given initial conditions. Determining this involves understanding how the solutions to the differential equation behave near the initial condition and how these solutions' derivatives reflect changes in slope and concavity.

Understanding when and why a function has an extremum can be pivotal, especially in fields such as physics and engineering, where it is critical to know the points at which a system reaches its peak or lowest state. This problem not only tests your ability to apply derivative tests in the context of differential equations but also reinforces the connection between the abstract mathematical concepts and their practical application in modeling and predicting real-world phenomena.