Find the maximum or minimum values, if any, of the following funcitons without using the derivatives :
`f(x)=-(x-1)^(2)+10`

Answer

To find the maximum or minimum values of the function \( f(x) = -(x-1)^2 + 10 \) without using derivatives, we can analyze the function's structure.

### Step-by-Step Solution:

1. **Identify the Structure of the Function**:
The function can be rewritten as:
\[ f(x) = -1 \cdot (x-1)^2 + 10 \]
Here, \((x-1)^2\) is a squared term, which is always non-negative (i.e., \((x-1)^2 \geq 0\) for all \(x\)).

2. **Determine the Maximum Value**:
Since \((x-1)^2\) is always non-negative, the maximum value of \(-(x-1)^2\) occurs when \((x-1)^2\) is at its minimum, which is 0. This happens when:
\[ x - 1 = 0 \implies x = 1 \]
Substituting \(x = 1\) into the function gives:
\[ f(1) = -0 + 10 = 10 \]
Thus, the maximum value of \(f(x)\) is 10.

3. **Determine the Minimum Value**:
As \(x\) moves away from 1 (either increasing or decreasing), \((x-1)^2\) becomes positive, leading to \(-(x-1)^2\) becoming negative. Therefore, as \(x\) approaches positive or negative infinity, \(f(x)\) will approach:
\[ f(x) \to -\infty \]
This indicates that there is no minimum value for the function.

### Conclusion:
- The maximum value of the function \( f(x) = -(x-1)^2 + 10 \) is **10** at \( x = 1 \).
- There is **no minimum value** as \( f(x) \) approaches \(-\infty\).