Question

divide 24 into two parts such that the product of both the parts is 128​

Answer 1

Answer:

Step-by-step explanation:

Let the two parts be x and (24-x).

Their product =x(24-x)

Method 1

x(24-x)=24x - x^2 is maximum at x=-24/-2 =12.

{ a quadratic expression ax^2 +bx+c has max/min (max if a<0 and min if a>0) value

at x=-b/2a,since the curve is symmetrical about that line.

Obviously, here a=-1 and b=24.}

So,maximum product =12(24-12)=12×12=144.Ans.

Method 2

Use the relation,

A.M.>=G.M. on the entities x and (24-x).

=> x+(24-x) / 2 >= { x(24-x) } ^1/2

=> 12>={ x(24-x) } ^1/2

=> x(24-x) <= 12^2 =144.Ans.

Method 3

<Calculus>

Let A=x(24-x)

Differentiating A wrt x both sides,we get

dA/dx =24-x-x=24-2x =0 at x=12.

d^2A/dx^2 =-2 < 0.

Hence A is max at x=12.

And,Max A = 12(24-12)=12×12=144.Ans.