If the minimum value of`x^2+2x+3` is `m` and maximum value of `-x^2+4x+6` is M then `m+M=`

Options

A. 10
B. 11
C. 12
D. 13

Answer

To solve the problem, we need to find the minimum value of the function \( f(x) = x^2 + 2x + 3 \) and the maximum value of the function \( g(x) = -x^2 + 4x + 6 \). Finally, we will sum these two values.

### Step 1: Find the minimum value of \( f(x) = x^2 + 2x + 3 \)

1. Rewrite the quadratic in perfect square form:
\[ f(x) = x^2 + 2x + 3 = (x^2 + 2x + 1) + 2 = (x + 1)^2 + 2 \]

2. The expression \( (x + 1)^2 \) is always non-negative and achieves its minimum value of 0 when \( x = -1 \).

3. Therefore, the minimum value of \( f(x) \) is:
\[ m = (0) + 2 = 2 \]

### Step 2: Find the maximum value of \( g(x) = -x^2 + 4x + 6 \)

1. Rewrite the quadratic in standard form:
\[ g(x) = -x^2 + 4x + 6 \]
We can factor out -1 from the first two terms:
\[ g(x) = - (x^2 - 4x) + 6 \]

2. Complete the square for \( x^2 - 4x \):
\[ x^2 - 4x = (x^2 - 4x + 4) - 4 = (x - 2)^2 - 4 \]
Thus,
\[ g(x) = -((x - 2)^2 - 4) + 6 = - (x - 2)^2 + 4 + 6 = - (x - 2)^2 + 10 \]

3. The expression \( - (x - 2)^2 \) is always non-positive and achieves its maximum value of 0 when \( x = 2 \).

4. Therefore, the maximum value of \( g(x) \) is:
\[ M = 0 + 10 = 10 \]

### Step 3: Calculate \( m + M \)

Now, we can find the sum of the minimum and maximum values:
\[ m + M = 2 + 10 = 12 \]

### Final Answer:
\[ \boxed{12} \]