The maximum value `49-(7-x)^2` is

Options

A. 49
B. 42
C. 0
D. none of these

Answer

To find the maximum value of the function \( f(x) = 49 - (7 - x)^2 \), we can follow these steps:

### Step 1: Identify the function
We start with the function:
\[ f(x) = 49 - (7 - x)^2 \]

### Step 2: Differentiate the function
To find the maximum value, we need to differentiate the function with respect to \( x \):
\[ f'(x) = \frac{d}{dx}[49 - (7 - x)^2] \]
Using the chain rule, we get:
\[ f'(x) = 0 - 2(7 - x)(-1) = 2(7 - x) \]

### Step 3: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:
\[ 2(7 - x) = 0 \]
Solving for \( x \):
\[ 7 - x = 0 \implies x = 7 \]

### Step 4: Determine if it is a maximum or minimum
Next, we need to check whether this critical point is a maximum or minimum by finding the second derivative:
\[ f''(x) = \frac{d}{dx}[2(7 - x)] = -2 \]
Since \( f''(x) < 0 \), this indicates that the function has a maximum at \( x = 7 \).

### Step 5: Calculate the maximum value
Now, we substitute \( x = 7 \) back into the original function to find the maximum value:
\[ f(7) = 49 - (7 - 7)^2 = 49 - 0 = 49 \]

### Conclusion
Thus, the maximum value of the function \( 49 - (7 - x)^2 \) is:
\[ \boxed{49} \]

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