Question

Find two positive numbers x and y such that their sum is 35 and the product x^ 2 y^ 5 is a maximum

Answer 1

HELLO DEAR,

two positive no. x and y and there sum if 35
it means x + y = 35 => y = (35 - x).

and let f(x) = x²y^5

then, f(x) = x²(35 - x)^5

f'(x) = 5x²(35 - x)⁴ ( - 1) + (35 - x)^5 * 2x

f'(x) = x(35 - x)⁴(-5x + 70 - 2x)

f'(x) = x(35 - x)⁴(70 - 7x)

now, f'(x) = 0

x(35 - x)⁴(70 - 7x) = 0

x = 0 , x = 35 , x = 10

When x = 35, and y = 35 − 35 = 0.
and the product x² y^5 equal to 0.

When x = 0, and y = 35 − 0 = 35
and the product x²y^5 will be 0.

∴ x = 0 and y = 35 cannot be the possible values.

hence, Only critical point is  x = 10

now, f"(x) = d(35 - x)⁴dx * (70x - 7x²) + d(70x - 7x²)/dx * (35 - x)⁴

f"(x) = (-4)(70x - 7x²)*(35 - x)³ + (70 - 14x)*(35 - x)⁴

f"(x) at x = 10,

f"(x) = -4(700 - 700)(35 - 10)³ + (70 - 140)(35 - 10)⁴

f"(x) = 0 -70*(25)⁴ < 0

then , f"(x) < 0 whene x = 10
=> p is maximum when x = 10

thus, when x = 10,

y = 35 - 10

y = 25.

hence,x = 10, y = 25


I HOPE ITS HELP YOU DEAR,
THANKS