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

Given the function:

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

Simplifying:

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

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

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

Combining like terms:

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

Step 1: Finding Maximum and Minimum Values

To find the maximum and minimum values of \(f(x)\), we differentiate with respect to \(x\):

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

Since the derivative is constant and negative, \(f(x)\) is a linear function that decreases as \(x\) increases. Therefore, the maximum value occurs at the lower bound of the domain of \(x\), and the minimum value occurs at the upper bound.

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

For the square root to be real, we require:

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

Solving the quadratic inequality:

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

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

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

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

Thus, the minimum value \(m = 9\).

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

Given that \(M = 49\):

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

Therefore, the correct answer is 1600.