Question

Find the largest positive value attained by the function $\begin{aligned} f(x)&=\sqrt{8x-x^2}\\ &\quad-\sqrt{14x-x^2-48}, \end{aligned}$ $x$ a real number.

$\sqrt7-1$

$3$

$2\sqrt3$

$4$

$\sqrt{55}-\sqrt5$

Answer: C

Concepts:radical function domain maximizing expression

Difficulty rating: 2080

Small Hint:

Factor both radicands using the common factor $8-x$

Big Hint:

On the domain $6\le x\le8,$ rationalize $\sqrt{8-x}(\sqrt x-\sqrt{x-6})$

Solution:

Both radicals are real exactly when $6\le x\le8.$ Factoring and rationalizing, $\begin{aligned} f(x) &=\sqrt{8-x}\bigl(\sqrt x-\sqrt{x-6}\bigr)\\ &=\frac{6\sqrt{8-x}}{\sqrt x+\sqrt{x-6}}. \end{aligned}$ On this interval the numerator decreases while the denominator increases, so the maximum occurs at $x=6.$ Its value is $f(6)=\sqrt{12}=2\sqrt3.$ Thus the correct answer is C.

1993 AMC 12 Problem 26

Answer: C

Difficulty rating: 2080

Small Hint:

Big Hint:

Solution:

Problem 26 in Other Years