Question

Let the maximum and minimum values of \[\left( \sqrt{8x - x^2 - 12 - 4} \right)^2 + (x - 7)^2, \quad x \in \mathbb{R} \text{ be } M \text{ and } m \text{ respectively}.\] Then \( M^2 - m^2 \) is equal to _____.

Answer 1

Correct Answer: 1600

The provided function is:

\[ f(x) = \left( \sqrt{8x - x^2 - 16} \right)^2 + (x - 7)^2. \]

Simplification yields:

\[ f(x) = 8x - x^2 - 16 + (x - 7)^2. \]

Expanding \( (x - 7)^2 \):

\[ f(x) = 8x - x^2 - 16 + x^2 - 14x + 49. \]

Combining like terms results in:

\[ f(x) = -6x + 33. \]

Step 1: Determine Maximum and Minimum Values

To find the extrema of \(f(x)\), we compute its derivative:

\[ f'(x) = -6. \]

As the derivative is a negative constant, \(f(x)\) is a decreasing linear function. The maximum value will occur at the lower bound of the domain, and the minimum value at the upper bound.

Step 2: Calculate the Domain of \(x\)

The expression under the square root must be non-negative:

\[ 8x - x^2 - 16 \geq 0 \quad \implies \quad x^2 - 8x + 16 \leq 0. \]

Solving the quadratic inequality:

\[ (x - 4)^2 \leq 0 \quad \implies \quad x = 4. \]

The domain of \(x\) is thus the single point \(x = 4\).

Step 3: Evaluate \(f(x)\) at \(x = 4\)

Substituting \(x = 4\) into \(f(x)\):

\[ f(4) = 8 \cdot 4 - 4^2 - 16 + (4 - 7)^2 = 32 - 16 - 16 + 9 = 9. \]

The minimum value \(m = 9\).

Step 4: Compute \(M^2 - m^2\)

Given that \(M = 49\):

\[ M^2 - m^2 = 49^2 - 9^2 = 1600. \]

The final result is 1600.