Question

the two positive numbers whose sum is 12 and sum of the squares in minimum are
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Answer 1

Answer: The numbers are 7.5 and 7.5.

Step-by-step explanation:

Let the first number is x,

Since, the sum of two number is 15,

⇒ First number + second number = 15,

⇒ x + second number = 15

⇒ Second number = 15 - x

Let f(x) shows the sum of the squares of the number,

⇒ f(x) = x² + (15-x)² = 2x²- 30x + 225,

By differentiating with respect to x,

We get,

f'(x) = 4x - 30,

For maximum or minimum, f'(x) = 0,

⇒ 4x - 30 = 0 ⇒ x = 7.5,

Again differentiating f'(x) with respect to x,

f''(x) = 4

At x = 7.5 f''(x) = Positive,

Thus, f(x) is minimum at x = 7.5,

Hence, the first number is 7.5,

And, the second number is 15 - 7.5 = 7.5

Answer 2

Answer:

x=6 and y=6

Step-by-step explanation:

x+y=12

y=12-x--(1)

f(x) =x^2+y^2--(2)

substitute (1) in (2)

f(x)=x^2+(12-x)^2

=x^2+x^2+144-24x

=2x^2-24x+144

f(x)=x^2-12x+72--(3)

diffrentiate equation (3)

f'(x) =2x-12

f'(x) =0

2x-12=0

2x=12

x=6

substitute x=6 in (1)

y=12-6

y=6

therefore, x=6 and y=6