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)

Answer 1

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 \[ y= \frac 4 {x}= 2 \sqrt 2\]