As a high school student passionate about mathematics, I recently came across an interesting idea that could provide a useful shortcut when identifying the relative extrema of a function. A theorem that builds on the standard second derivative test, and I think it's worth sharing with the wider mathematical community.
The Standard Second Derivative Test
The classic approach to determining relative extrema involves first finding the critical points of a function by setting the first derivative equal to zero. Then, we apply the second derivative test:
- If f''(x) > 0 at a critical point x, then f(x) has a relative minimum at x.
- If f''(x) < 0 at a critical point x, then f(x) has a relative maximum at x.
- If f''(x) = 0 at a critical point x, the test is inconclusive.
This method generally works well, but I noticed that there are some scenarios where the second derivative test falls short. That's where my theorem comes in.
Soham's Theorem
The critical points of a function f(x) are only relative extrema if the first derivative f'(x) does not have a relative extrema at the same critical point. In other words:
The critical number of f(x) is a relative extrema if f'(x) does not have a relative extrema at the same critical number as f(x).
This claim seems to hold true across the examples Soham has explored, providing a useful supplement to the standard second derivative test. Let's walk through one of his examples to see how it works in practice.
Applying Soham's Theorem
Consider the function f(x) = x^4 - 8x^3 + 6. To find the critical points, we set the first derivative equal to zero:
f'(x) = 4x^3 - 24x = 0 x = 0 and x = 6
Now, we need to check if these critical points are relative extrema. Using Soham's theorem, we look at the first derivative f'(x) = 4x^3 - 24x and find its critical points by setting the second derivative equal to zero:
f''(x) = 12x^2 - 24 = 0 x = 0 and x = 2
We see that the critical point x = 0 of the original function f(x) corresponds to a critical point of the derivative f'(x), so by Soham's theorem, x = 0 cannot be a relative extremum of f(x).
However, the critical point x = 6 of f(x) does not correspond to a critical point of f'(x), so x = 6 must be a relative extremum of f(x). This matches the results we would get from the standard second derivative test.
Limitations and Future Work
Soham believes that his theorem may have the same limitations as the second derivative test, namely that it cannot be applied to exponential or logarithmic functions. He is still exploring the boundaries of this new approach.
Additionally, Soham points out an interesting paradox that emerges when applying his theorem - to determine the relative extrema of one function, you end up needing to calculate the relative extrema of another function. This recursive nature is an intriguing aspect that warrants further investigation.
I'm excited to see where Soham takes this idea. His clever thinking and dedication to understanding the nuances of relative extrema are inspiring. I encourage the mathematical community to engage with his work and provide feedback. Who knows - Soham's theorem could become a valuable tool in the toolbox of calculus students and enthusiasts alike.