Find two positive real numbers with the given sum whose product is a maximum. the sum of the first and twice the second is 24

Y+2x=24
product is max
xy=max

we have
y+2x=24
minus 2x both sides
y=24-2x
sub that for y in xy=max
(24-2x)(x)=max
-2x^2+24x=max
we have a parabola opening down
the coordinates of the vertex is our answer

the x value of the vertex is -b/2a where
ax^2+bx+c=y

-24/2(-2)=-24/-4=6
xvalue=6

we can just sub
y=24-2x
y=24-2(6)
y=24-12
y=12
the numbers are 6 and 12
the product is 72

the numbers are 6 and 12

the formula of a slope:

from a table we have

substitute:

the slope of f(x) is -1.

the graph of funcfion g(x) is a horizontal line. the slope of a horizontal line is equal 0. therefore the slope of g(x) is 0.

-1 < 0

answer: the function g(x) has the greatest slope.