Application of Derivatives · Class 12 · CBSE · Science Mathematics
Two positive numbers have sum 20. Find the maximum possible value of their product.
Practice question from Application of Derivatives (Class 12 · CBSE · Science Mathematics), with the correct answer and a step-by-step solution.
medium · Subjective · Q-C48718
Two positive numbers have sum 20. Find the maximum possible value of their product.
Answer: Let the numbers be x and 20 − x, so their product is P(x) = x(20 − x) = 20x − x². Since P′(x) = 20 − 2x, the critical point is x = 10 and P″(x) = −2 < 0, so the product is maximum when both numbers are 10; the maximum product is 100.
Solution
Let one number be x; the other is 20 − x. The product is P(x) = 20x − x², whose derivative is zero at x = 10 and whose second derivative is negative, so the maximum product is 10 × 10 = 100.
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