2024학년도 (2023년 시행) 고3 5월(4월) 모의고사 공통 22번

Have you ever danced with numbers and equations, waltzing through the intricate world of mathematics? Well, put on your mathematical dancing shoes, because we're about to embark on a journey into the fascinating realm of functions, where gx takes the stage, and the spotlight shines on the enigmatic g'x.

The Prologue: A Prelude to gx and fx

Our story begins with a curious duo, gx and fx. Picture gx as the leading character, a function that's not just any ordinary function. It's differentiable, and its derivative, g'x, boasts the mark of continuity. But wait, the plot thickens as we are presented with a challenge: we need to determine g'x. Where do we start?

Setting the Stage: Seeking Key Points

Let's start our quest by examining the boundaries, where the stage is set between -2 and 2. As we delve into the mathematics of this theatrical performance, we realize that when we introduce the same number at both extremes, the result is an enchanting number: zero. Yes, g(0) is zero, and we'll keep that as our first treasure.

Now, let's move on to the exhilarating act of differentiation. As we take a step into the world of calculus, we find ourselves differentiating gx. With x in the leading role, the performance unveils itself as -x + a. But, there's a twist in the plot. We need to pay attention to the boundaries once more. When x is less than -2 or greater than or equal to 2, g'x remains a mystery.

Act 1: The Graph Unveiled

With our mathematical quill in hand, let's sketch the first act of our graph. As we navigate the range from -2 to 2, a remarkable linear graph takes shape: -x + 1. It gracefully descends, creating a peak at 1 and gifting us with a local maximum.

Act 2: The Art of Transformation

Now, our protagonist, g'x, undergoes a transformation. It twists and turns, creating a vivid V-shape as it rises and falls. This transformation is key to achieving the coveted status of a quadratic function. The audience awaits in anticipation as g'x evolves.

Act 3: A Twist in the Tale

But the story doesn't end there. Act 3 brings a surprise twist. The second point of interest, g'(1), should also be zero to maintain the integrity of our quadratic function. As the spotlight shines on this point, we discover that it must be flat as a pancake. This gives us another piece of the puzzle, and we set g'(1) to zero.

Act 4: The Grand Finale

With the pieces of the puzzle falling into place, we approach the grand finale. The point where g'x pierces the x-axis is crucial. This is where the magic happens – our quadratic function emerges in all its glory. By understanding the art of transformation and the symmetrical nature of functions, we can determine that g'x must gracefully pierce the x-axis twice.

The Epilogue: Unveiling the Mysteries

As the curtains draw to a close on our mathematical performance, we've unraveled the mysteries of gx and g'x. The captivating dance between functions and derivatives has led us to the grand finale: g'(1) = 0 and the elegant quadratic form of our function.

In the realm of mathematics, the beauty lies not only in the numbers but also in the intricate choreography of functions. So, next time you encounter a mathematical puzzle, remember that it might just be a dance waiting to be unveiled, with gx and g'x taking center stage, creating a symphony of numbers and equations that leaves you in awe of the mathematical world.