Finding the Minimum Value of the Function
We need to find the minimum value of the function $f(x) = x^{2025} - x^{2000}$ on the interval $x \in [0, 1]$.
Step 1: Calculate the Derivative
First, find the derivative of $f(x)$ with respect to $x$.
$ f'(x) = \frac{d}{dx}(x^{2025} - x^{2000}) = 2025 x^{2024} - 2000 x^{1999} $
Step 2: Find Critical Points
Set the derivative equal to zero to find the critical points:
$ 2025 x^{2024} - 2000 x^{1999} = 0 $
Factor out the common term $x^{1999}$:
$ x^{1999} (2025 x^{25} - 2000) = 0 $
This yields two possibilities:
- $x^{1999} = 0 \implies x = 0$
- $2025 x^{25} - 2000 = 0 \implies 2025 x^{25} = 2000 \implies x^{25} = \frac{2000}{2025} = \frac{80}{81}
For the second case, $x = \left(\frac{80}{81}\right)^{\frac{1}{25}}$. This value lies within the interval $[0, 1]$. Let $x_c = \left(\frac{80}{81}\right)^{\frac{1}{25}}$.
Step 3: Evaluate the Function at Endpoints and Critical Points
Evaluate $f(x)$ at the endpoints ($x=0, x=1$) and the critical point ($x_c$).
- At $x=0$: $f(0) = 0^{2025} - 0^{2000} = 0$.
- At $x=1$: $f(1) = 1^{2025} - 1^{2000} = 1 - 1 = 0$.
- At $x_c = \left(\frac{80}{81}\right)^{\frac{1}{25}}$:
$ f(x_c) = \left(\left(\frac{80}{81}\right)^{\frac{1}{25}}\right)^{2025} - \left(\left(\frac{80}{81}\right)^{\frac{1}{25}}\right)^{2000} $
$ f(x_c) = \left(\frac{80}{81}\right)^{81} - \left(\frac{80}{81}\right)^{80} $
Factor out $\left(\frac{80}{81}\right)^{80}$:
$ f(x_c) = \left(\frac{80}{81}\right)^{80} \left(\frac{80}{81} - 1\right) $
$ f(x_c) = \left(\frac{80}{81}\right)^{80} \left(\frac{-1}{81}\right) $
$ f(x_c) = \frac{-1}{81} \left(\frac{80}{81}\right)^{80} = (-81)^{-1} \cdot 80^{80} \cdot 81^{-80} = 80^{80} \cdot (-81)^{-81} $
Step 4: Determine the Minimum Value
Comparing the values $f(0)=0$, $f(1)=0$, and $f(x_c) = 80^{80} (-81)^{-81}$.
Since $x \in [0, 1]$, $x^{2025} \le x^{2000}$, which means $f(x) = x^{2025} - x^{2000} \le 0$. The minimum value must be negative.
Therefore, the minimum value is $f(x_c) = 80^{80} (-81)^{-81}$.
Step 5: Solve for n
The problem states that the minimum value is equal to $(80)^{80} (n)^{-81}$.
Equating the calculated minimum value with the given form:
$ 80^{80} (-81)^{-81} = (80)^{80} (n)^{-81} $
Divide both sides by $80^{80}$:
$ (-81)^{-81} = n^{-81} $
This implies:
$ n = -81 $
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