Q. Let `f(x)=10-|x-10|` `AA`x`in[-9,9]` If M and m be the maximum and minimum value of f(x) respectively then

Options

A. `M +m =0`
B. `2M +m =-9`
C. `2M+m =7`
D. `M =m =7`

Answer

To solve the problem, we need to analyze the function \( f(x) = 10 - |x - 10| \) for \( x \) in the interval \([-9, 9]\).

### Step 1: Understanding the Absolute Value
The function involves an absolute value, which we need to break down based on the definition of absolute value. The expression \( |x - 10| \) can be rewritten as:
- \( -(x - 10) = 10 - x \) when \( x < 10 \)
- \( x - 10 \) when \( x \geq 10 \)

Since our interval is \([-9, 9]\), we only need to consider the case where \( x < 10 \).

### Step 2: Rewrite the Function
For \( x \) in the interval \([-9, 9]\):
\[ f(x) = 10 - |x - 10| = 10 - (10 - x) = x \]
Thus, in the interval \([-9, 9]\), the function simplifies to:
\[ f(x) = x \]

### Step 3: Determine the Maximum and Minimum Values
Since \( f(x) = x \) is a linear function, it will attain its maximum and minimum values at the endpoints of the interval \([-9, 9]\).

- Minimum value occurs at \( x = -9 \):
\[ f(-9) = -9 \]

- Maximum value occurs at \( x = 9 \):
\[ f(9) = 9 \]

### Step 4: Conclusion
Let \( M \) be the maximum value and \( m \) be the minimum value of \( f(x) \):
- \( M = 9 \)
- \( m = -9 \)

Thus, the maximum value \( M \) is 9 and the minimum value \( m \) is -9.

### Final Answer
The maximum value \( M \) is 9 and the minimum value \( m \) is -9.

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