Question 52 of 82intermediate🔧 ApplyNumerical4 marks

Find two positive numbers x and y such that their sum is 35 and the product x²y⁵ is a maximum.

Correct Answer

The two positive numbers are x = 10 and y = 25.

Exercise: EXERCISE 6.3 | Q: 15 | (Page 30)
For More Understanding

Explanation

This question involves finding maximum of a function using derivatives. The student needs to express one variable in terms of the other using the constraint equation, then differentiate and find critical points. The logarithmic differentiation method simplifies the calculation significantly for products with powers.

Solution Steps

  1. Step 1: Let the two positive numbers be x and y. Given: x + y = 35, and we need to maximize P = x²y⁵.

  2. Step 2: From x + y = 35, we get y = 35 - x. Substituting in P: P = x²(35 - x)⁵

  3. Step 3: Taking logarithm on both sides: ln P = 2 ln x + 5 ln(35 - x)

  4. Step 4: Differentiating with respect to x: (1/P)(dP/dx) = 2/x - 5/(35 - x)

  5. Step 5: For maximum, dP/dx = 0. Therefore: 2/x - 5/(35 - x) = 0

  6. Step 6: Solving: 2(35 - x) = 5x, which gives 70 - 2x = 5x, so 7x = 70, hence x = 10

  7. Step 7: Therefore, y = 35 - 10 = 25

  8. Step 8: Verification using second derivative test: d²P/dx² at x = 10 is negative, confirming maximum. Hence, the two numbers are x = 10 and y = 25.

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