The minimum and maximum values of `(7-x)^2-8` are

Options

A. `-8,15`
B. `-15,15`
C. `-8,41`
D. none of these

Answer

To find the minimum and maximum values of the function \( f(x) = (7 - x)^2 - 8 \), we will follow these steps:

### Step 1: Identify the function
We start with the function:
\[ f(x) = (7 - x)^2 - 8 \]

### Step 2: Differentiate the function
Next, we differentiate \( f(x) \) with respect to \( x \) to find the critical points:
\[ f'(x) = \frac{d}{dx}((7 - x)^2 - 8) \]
Using the chain rule:
\[ f'(x) = 2(7 - x)(-1) = -2(7 - x) = 2(x - 7) \]

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

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

### Step 5: Calculate the minimum value
Now, we substitute \( x = 7 \) back into the original function to find the minimum value:
\[ f(7) = (7 - 7)^2 - 8 = 0 - 8 = -8 \]

### Step 6: Determine the maximum value
As \( x \) increases or decreases away from 7, \( (7 - x)^2 \) will increase indefinitely. Therefore, the function does not have a maximum value; it approaches infinity as \( x \) moves away from 7.

### Final Answer
- Minimum value: \(-8\)
- Maximum value: Not defined (approaches infinity)