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Tìm giá trị lớn nhất của Vx^2-2x+5 -10

To find the maximum value of the given function:

\[
C = \sqrt{x^2 - 2x + 5} - 10
\]

First, we analyze the function step-by-step.

### Step 1: Rewrite the quadratic expression inside the square root.
We can complete the square for \( x^2 - 2x + 5 \):

\[
x^2 - 2x + 5 = (x - 1)^2 + 4
\]

Thus, the equation becomes:

\[
C = \sqrt{(x - 1)^2 + 4} - 10
\]

### Step 2: Analyze the expression inside the square root.
Since \( (x - 1)^2 \) is always non-negative, the smallest value of \( (x - 1)^2 \) is 0, which occurs when \( x = 1 \). Therefore, the minimum value of \( \sqrt{(x - 1)^2 + 4} \) is:

\[
\sqrt{0 + 4} = \sqrt{4} = 2
\]

As \( (x - 1)^2 \) increases, \( \sqrt{(x - 1)^2 + 4} \) also increases. Hence, the smallest value of the expression is 2, and the expression grows larger as \( x \) moves away from 1.

### Step 3: Maximum of \( C \)
The smallest value of \( \sqrt{(x - 1)^2 + 4} \) is 2, which means the maximum value of \( C \) is:

\[
C_{\text{max}} = 2 - 10 = -8
\]

Thus, the maximum value of \( C \) is \(-8\), and it occurs when \( x = 1 \).

### Conclusion:
The maximum value of \( C \) is \(-8\), and it happens at \( x = 1 \).

Would you like more details on any specific step or have any questions?

### Related Questions:
1. How do you complete the square for a quadratic expression?
2. Can you explain how to determine the range of a square root function?
3. What are the properties of quadratic functions that affect their maxima or minima?
4. How does shifting a function horizontally or vertically affect its graph?
5. What are the key steps in finding the vertex of a parabola?

### Tip:
Always check the domain of the function when dealing with square roots to ensure you're considering only valid values for which the expression inside the root is non-negative.

Tìm giá trị lớn nhất của C = \sqrt{x^2 - 2x + 5} - 10

This problem explores how to find the maximum value of the function C = sqrt(x^2 - 2x + 5) - 10 using the technique of completing the square. Ideal for students learning advanced algebra concepts, including quadratic functions and properties of square roots.

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