Question 50 of 82intermediate🔧 ApplyNumerical4 marks

Find two numbers whose sum is 24 and whose product is as large as possible.

Correct Answer

The two numbers are 12 and 12, with maximum product 144.

Exercise: EXERCISE 6.3 | Q: 13 | (Page 30)
For More Understanding

Explanation

This is an optimization problem using derivatives. We express the product as a function of one variable using the constraint, then find the critical point where the derivative equals zero. The second derivative test confirms it's a maximum.

Solution Steps

  1. Step 1: Let the two numbers be x and y. Given that x + y = 24, so y = 24 - x.

  2. Step 2: Let P denote the product of the two numbers. Then P = x · y = x(24 - x) = 24x - x².

  3. Step 3: Differentiating P with respect to x: P'(x) = 24 - 2x.

  4. Step 4: For maximum product, set P'(x) = 0. Thus, 24 - 2x = 0, which gives x = 12.

  5. Step 5: Verify maximum using second derivative: P''(x) = -2 < 0. Since P''(x) < 0, the product is maximum at x = 12.

  6. Step 6: Therefore, y = 24 - 12 = 12. The two numbers are 12 and 12, and the maximum product is 144.

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