Exploring the Minimum Value of a Function Through Differentiation Techniques

Understanding the minimum value of a function is a fundamental concept in calculus that helps us analyze the behavior of mathematical functions. In this article, we will delve into the process of finding the minimum value of the function f(x) x^2 frac{1}{x^2 1} - 3. We will also verify the derivative and use the First Derivative Test to confirm the minimum value.

Rewriting the Function for Analysis

To begin, let's rewrite the function for easier analysis:

f(x) x^2 - 3 frac{1}{x^2 1}

Finding Critical Points

To find the critical points, we need to take the derivative of f(x) and set it to zero. Let's start by computing the first derivative:

f'(x) 2x - frac{1 cdot 2x}{(x^2 1)^2} 2x - frac{2x}{(x^2 1)^2}

Setting f'(x) 0, we get:

2x left(1 - frac{1}{(x^2 1)^2} right) 0

This equation has two cases:

2x 0 implies x 0

1 - frac{1}{(x^2 1)^2} 0 implies (x^2 1)^2 1

Solving for the second case:

Therefore, the only critical point is x 0.

Evaluating the Function at the Critical Point

Next, we evaluate the function at x 0:

f(0) 0^2 - 3 frac{1}{0^2 1} 0 - 3 -2

Behavior at Infinity and Second Derivative Test

To confirm if x 0 is a minimum, we need to check the behavior as x approaches positive and negative infinity. As x to pm infty:

f(x) approx x^2 - 3 to infty

Thus, f(x) approaches infinity as x increases. Now, let's check the second derivative to determine concavity:

f''(x) 2 - frac{2}{(x^2 1)^2} cdot 8x^2 2 - frac{16x^2}{(x^2 1)^3}

Evaluating at x 0:

f''(0) 2 - 0 2

Since f''(0) 0, the function is concave up at x 0, confirming a local minimum.

Alternative Verification Using the Square Root Form

As an alternative approach, we can transform the function to:

P x^2 - 3 left(frac{1}{sqrt{x^2 1}} - 1 right)^2

The minimum value of the square term is 0 when x^2 ! 1, implying x 0.

Substituting x 0 into the function:

P_{min} -2

This confirms that the minimum value of the function is -2.

Conclusion

In conclusion, we have demonstrated that the minimum value of the function f(x) x^2 frac{1}{x^2 1} - 3 is -2. We used the First Derivative Test and verified the result using a square root transformation. This method provides a clear and systematic approach to finding the minimum value of complex functions.