Let's derivate #1

Using AM-GM

$\large{2x+y\ge 2\sqrt { 2xy } \\ { \left( \frac { 7 }{ 2 } \right) }^{ 2 }\ge 2xy\\ \frac { 49 }{ 8 } \ge xy}$

Maximum Value of $xy$ is $6.125$ .

2x + y = 7 ---> y= 7 - 2x . We are looking for a maximum for f(x)=x(7-2x)= 7x - 2(x^2) ; f '(x)=7 - 4x = 0 if and only if x=7/4. f ''(x)=f '' (7/4)= -4<0 for every x in R. This implies that f(x) has a relative maximum for x=7/4. The graph of f(x)=7x - 2x^2 is a parabola, then this implies that the relative maximum is an absolute maximum. Then f(7/4)= 49 /8 is the maximum of xy

Mr Templado's approach couldn't be much easier, if you use symbolic algebra and if you note that the function to be maximised must be a concave-upward parabola ( $xy$ is 0 where the line crosses the two axes).

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>>> from sympy import *
>>> var('x y')
(x, y)
>>> solve(2*x+y-7,y)
[-2*x + 7]
>>> diff(x*(-2*x + 7))
-4*x + 7
>>> solve(-4*x + 7)
[7/4]
>>> (x*(-2*x + 7)).subs(x,Rational(7,4))
49/8
>>> N(Rational(49,8))
6.12500000000000