Question

If 2x+y=10, find the minimum value of x^2+y^2?

Answer 1

`x^2 + y^2 = (4)^2 + 2^2 = 20`

Explanation:

Given:

`2x +y = 10`

We can rewrite in terms of `y` as

`y = 10 - 2x`

Now substitute

`x^2 + y^2`

`x^2 + (10 -2x)^2`

`x^2 + 100 - 40x + 4x^2`

Call the value of this expression `A`.

`A = 5x^2 - 40x + 100`

We notice that `A` is a parabola that opens upwards. So to find our minimum we can use differentiation.

`A' = 10x - 40`

The minimum will occur when the derivative equals `0`.

`0 = 10x - 40`

`0 = 10(x - 4)`

`x= 4`

Therefore, the minimum value will occur when `y = 2` and `x = 4`. This means the minimum value is `2^2 + 4^2 = 20`

Hopefully this helps!