I) The maximum value of `c + 2bx -x^(2)` is `c+b^(2)`
II) The minimum value of `x^(2) + 2bx + c` is `c-b^(2)`
Which of the above statements is true ?

Options

A. only I
B. only II
C. both I and II
D. neither I nor II

Answer

To determine the truth of the statements given, we will analyze each statement step by step.

### Step 1: Analyze Statement I
We need to find the maximum value of the expression \( c + 2bx - x^2 \).

1. **Rewrite the expression**:
\[ f(x) = c + 2bx - x^2 \]
Rearranging gives:
\[ f(x) = -x^2 + 2bx + c \]

2. **Complete the square**:
To complete the square, we can rewrite the quadratic part:
\[ f(x) = -\left(x^2 - 2bx\right) + c \]
Now, complete the square for \( x^2 - 2bx \):
\[ x^2 - 2bx = (x - b)^2 - b^2 \]
Substituting this back gives:
\[ f(x) = -\left((x - b)^2 - b^2\right) + c = - (x - b)^2 + b^2 + c \]

3. **Find the maximum value**:
The term \(-(x - b)^2\) reaches its maximum value of \(0\) when \(x = b\). Therefore, the maximum value of \(f(x)\) is:
\[ f(b) = c + b^2 \]

So, the maximum value of \(c + 2bx - x^2\) is indeed \(c + b^2\).

### Step 2: Analyze Statement II
Now we will find the minimum value of the expression \( x^2 + 2bx + c \).

1. **Rewrite the expression**:
\[ g(x) = x^2 + 2bx + c \]

2. **Complete the square**:
We can rewrite the quadratic part:
\[ g(x) = (x^2 + 2bx) + c \]
Completing the square for \(x^2 + 2bx\):
\[ x^2 + 2bx = (x + b)^2 - b^2 \]
Substituting this back gives:
\[ g(x) = (x + b)^2 - b^2 + c \]

3. **Find the minimum value**:
The term \((x + b)^2\) reaches its minimum value of \(0\) when \(x = -b\). Therefore, the minimum value of \(g(x)\) is:
\[ g(-b) = c - b^2 \]

So, the minimum value of \(x^2 + 2bx + c\) is indeed \(c - b^2\).

### Conclusion
Both statements are true. Therefore, the correct answer is that both statements are true.

### Final Answer
Both statements are true.