Toán 10 Chuyên Đề 06 Bài 1 Hàm Số Và Đồ Thị Phần Luyện Tập Lần 01

Mathematics is a beautiful journey, a relentless quest for understanding the world through numbers, shapes, and functions. In our pursuit of mathematical enlightenment, we often encounter intriguing problems that challenge our intellect and ignite the fires of curiosity. Today, we embark on a thrilling adventure, a quest to find the elusive value of "m" that makes a function defined for all real numbers.

Setting the Stage

Our mathematical odyssey begins with a simple yet profound question: What conditions must be met for a function to be defined for all real numbers (R)? The answer lies in solving the inequality (x^2 + mx - \frac{1}{3} \geq 0). This inequality sets the stage for our exploration.

The Art of Completing the Square

To unravel this enigma, we employ the art of completing the square. We transform our inequality into the elegant form ((x + \frac{m}{2})^2 - (\frac{m^2}{4} + \frac{1}{3}) \geq 0). This not only simplifies our equation but also reveals the hidden beauty within.

The Conditions Unveiled

Now, let's dissect the components of our inequality. To ensure our function is defined for all real numbers, we have two conditions to satisfy:

1. ((x + \frac{m}{2})^2) must always be greater than or equal to 0.

2. (\frac{m^2}{4} + \frac{1}{3}) must be less than or equal to 0.

Unleashing the Quadratic

For our first condition to hold true, (\frac{m^2}{4}) must be greater than or equal to 0. In other words, (m^2 \geq 0) or (m^2 \geq -\frac{4}{3}).

The Final Puzzle

Now, let's tackle the second condition. We need to satisfy (\frac{m^2}{4} + \frac{1}{3} \leq 0). This leads us to the following journey:

1. Multiply both sides of the equation by 4 to get (m^2 \leq -\frac{4}{3} \cdot 4).

2. Simplify further to (m^2 \leq -\frac{16}{3}).

This is where the mystery unfolds. To satisfy this condition, "m" cannot be a real number because there is no real number whose square is a negative value.

The Grand Revelation

In the grand tapestry of mathematics, we have discovered the truth. There is no real number "m" that can make the function defined for all real numbers (R). The quest has concluded, and the revelation is profound.

In our pursuit of understanding mathematical functions, we have delved into the intricate dance of inequalities, squares, and conditions. We've uncovered the hidden intricacies that govern the domain of functions. This journey serves as a reminder of the beauty and complexity that mathematics offers, always ready to challenge and inspire our intellect.

So, embrace the mysteries, for in the world of mathematics, even the unattainable reveals its secrets in due time.