Find two positive numbers whose product is 4 and such that 4x+2y is a minimum.
a)2 and 2 b)(1/4) and 4 c) (1/2) and 8 d) sqrt of 3 and (4/sqrt of 3) e) sqrt of 2 and (2*sqrt of 2)

Question

Find two positive numbers whose product is 4 and such that 4x+2y is a minimum.
a)2 and 2 b)(1/4) and 4 c) (1/2) and 8 d) sqrt of 3 and (4/sqrt of 3) e) sqrt of 2 and (2*sqrt of 2)

anonymous · Answer

The answer is e)
$$ y = \frac 4 x\
f(x) = 4 x + 2y= 4 x + \frac 8 x\
f'(x)= 4-\frac{8}{x^2}\
f'(x)=0 \text { for}  x =\pm \sqrt 2\
f''( \sqrt 2) > 0\
$$
The minimum is at $$ x= \sqrt 2$$
and y will be $$ y= \frac 4 {x}= 2 \sqrt 2$$