The number 24 is divided into two parts the maximum value of the product of the two parts is

Divide the number 20 into two parts such that their product is maximum.

The given number is 20.

Let x be one part of the number and y be the other part.

∴ x + y = 20

∴ y = (20 - x)      ...(i)

The product of two numbers is xy.

∴ f(x) = xy = x(20 - x) = 20x - x2

∴ f'(x) = 20 - 2x  and  f''(x) = - 2

Consider, f '(x) = 0

∴ 20 - 2x = 0

∴ 20 = 2x

∴ x = 10

For x = 10,

f ''(10) = - 2 < 0

∴ f(x), i.e., product is maximum at x = 10

and 10 + y = 20      ....[from (i)]

i.e., y = 10.

Concept: Maxima and Minima

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Let that no. are x and 24-x. So A/c to question(x)(24-x)=0 Since max. Value will lie when dy/dx=0 Now x=(24x-x^2)

dy/dx=24-2x=0. Hence x=12 and max. Product value will be 144