Mathematics : Application of Derivatives

The point on the curve x² = 2y which is nearest to the point (0, 5) is

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• (0, 0)

• (0, 0)

Determine two positive numbers whose sum is 15 and the sum of whose, squares is minimum.

Amongst all pairs of positive numbers with product 64, find those whose sum is the least.

Determine two positive numbers whose sum is 30 and whose product is the maximum. 

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

Find two positive numbers x and y such that x + y = 60 and xy³ is maximum.

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

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