The sum of two nonnegative numbers is 36. Find the numbers if
a. the difference of their square roots is to be as large as possible.
b. the sum of their square roots is to be as large as possible.

검증된 단계별 안내

Step 1: Define the two nonnegative numbers as x and y, with the constraint x + y = 36.

Step 2: For part (a), express the difference of their square roots as |√x - √y|. To maximize this difference, consider the function f(x) = √x - √(36 - x) and find its critical points by taking the derivative and setting it to zero.

Step 3: Calculate the derivative of f(x) with respect to x, which is f'(x) = (1/(2√x)) - (1/(2√(36-x))). Set f'(x) = 0 to find the critical points.

Step 4: Solve the equation (1/(2√x)) = (1/(2√(36-x))) to find the value of x that maximizes the difference of square roots. Check the endpoints x = 0 and x = 36 to ensure the maximum is within the interval.

Step 5: For part (b), express the sum of their square roots as √x + √y. To maximize this sum, consider the function g(x) = √x + √(36 - x) and find its critical points by taking the derivative and setting it to zero. Follow similar steps as in part (a) to find the value of x that maximizes the sum of square roots.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function within a given domain. In this problem, we need to optimize the difference and sum of the square roots of two nonnegative numbers, subject to the constraint that their sum is 36. This typically involves using techniques such as derivatives to find critical points.

Constraints

Constraints are conditions that must be satisfied in an optimization problem. Here, the constraint is that the sum of the two nonnegative numbers equals 36. Understanding how to express one variable in terms of another using this constraint is crucial for setting up the optimization equations.

Square Roots

Square roots are mathematical functions that return a nonnegative number whose square equals the given number. In this context, we are interested in the properties of square roots to maximize their difference and sum. Recognizing how square roots behave, especially in relation to their inputs, is essential for solving the optimization problems presented.

The sum of two nonnegative numbers is 36. Find the numbers ifa. the difference of their square roots is to be as large as possible.b. the sum of their square roots is to be as large as possible.