Question

minimize x^2+y^2 if 2x+y=10

Answer 1

well as it is given that 2x + y = 10
=> y = 10 - 2x

use this value in y^2 of the equation x^2 + y^2

and in the final answer you would be left with variable x only

Answer 2

i mean the terms with variable x

Answer 3

@kyle2393 you there ... ?

Answer 4

yeah sorry i was trying to figure it out

Answer 5

does x=10?

Answer 6

\[f(x)=x^2+y^2=x^2+(10-x)^2\]
find f'(x) and put =0 and find the value of x
find f''(x) at this value of x
if it is >0 then f(x) is minimum and find f(x) at this value of x
if f''(x)<0 ,then we get maxima. we are not interested in it.

Answer 7

@kyle2393 ..... no x is not equal to 10..
just proceed with the equation that @surjithayer has written

Answer 8

just an anternative solution.
f(x) = x^2+ y^2 = x^2 + (10-2x)^2 = 5x^2 - 40x + 100
f(x) is a quadratic function which its kind is minimum (a>0, here a =5).
f(x) would be minimum when x = -b/(2a) = -(-40)/(2*5) = 4.
then to get the minimum value is just put x = 4 into f(x), in other words it is f(x=4) = 5(4)^2 - 40(4) + 100 = ...... ?

Answer 9

it can be solved like this also.
f(x)=5(x^2-8x+16-16)+100
=5(x^2-8x+16)-80+100
=5(x-4)^2+20
minimum value of f(x) is 20 and is attained at x=4