Understanding Stationary Points of the Function f(x) 1 - x^2

The concept of stationary points is fundamental in the realm of calculus, playing a pivotal role in understanding the behavior of functions. This article delves into the analysis of the function f(x) 1 - x^2, specifically focusing on finding and understanding its stationary points. Through this exploration, we will uncover key insights into the function's graph and its key features.

Introduction to Stationary Points

Stationary points, in the context of a function, are points where the derivative of the function is zero or undefined. These points are critical in calculus since they can indicate local maxima, minima, or points of inflection. The analysis of stationary points is essential for comprehending the behavior of functions and their graphical representations.

The Function in Focus: f(x) 1 - x^2

The function we will be analyzing is f(x) 1 - x^2. Let us break down the components of this function:

Graphical Depiction

Visually, the function f(x) 1 - x^2 represents a downward-opening parabola with its vertex at (0, 1). The parabola reaches its highest point at x 0 and declines symmetrically on either side, with the curve approaching the x-axis as x approaches positive or negative infinity.

Determining Stationary Points

To find the stationary points of the function, we first need to determine the derivative of f(x), denoted as f'(x).

Step-by-Step Derivatives

Let's start by computing the derivative of f(x) 1 - x^2.

Step 1: Apply the Power Rule

The power rule for differentiation states that if f(x) x^n, then f'(x) n * x^(n-1).

Applying this rule to our function, we get:

Let f(x) 1 - x^2. The derivative of the constant 1 is 0, and the derivative of -x^2 is -2x. Therefore, we have:

$$f'(x) 0 - 2x -2x$$

Step 2: Set the Derivative Equal to Zero

To find the stationary points, we set the derivative equal to zero:

$$-2x 0$$

Solving for x, we find:

$$x 0$$

Step 3: Verify the Nature of the Stationary Point

Since the derivative is a linear function, we can determine that the function f(x) 1 - x^2 has a single stationary point at x 0. The fact that the derivative changes sign from positive to negative around this point indicates that it is a local maximum.

Conclusion

The function f(x) 1 - x^2 has a single stationary point at x 0, which is a local maximum. This analysis provides valuable insights into the function's behavior, particularly its peak value at x 0, and helps in understanding the overall graphical representation of the function.

Further Exploration and Applications

The study of stationary points is not limited to theoretical exercises. This knowledge is applied in various fields, including economics, physics, and engineering. In economics, for example, stationary points can help in determining optimal production levels or pricing strategies. In physics, they can be used to analyze the positions of particles in motion.

For a deeper exploration, consider the following:

How does the function f(x) 1 - x^2 behave when subjected to second derivatives? Can you find and analyze the stationary points of other quadratic functions? What about higher-degree polynomial functions? How do their stationary points compare to those of quadratic functions?

By delving into these questions, one can gain a more comprehensive understanding of the role of stationary points in calculus.

Final Thoughts

The exploration of the stationary points of the function f(x) 1 - x^2 is a fundamental exercise in calculus. By understanding the behavior of the function around its stationary points, we can gain valuable insights into its graphical representation and its practical applications.