Lagrange multipliers:







(if

)

(if

)

(if

)
In the first octant, we assume

, so we can ignore the caveats above. Now,

so that the only critical point in the region of interest is (1, 2, 2), for which we get a maximum value of

.
We also need to check the boundary of the region, i.e. the intersection of

with the three coordinate axes. But in each case, we would end up setting at least one of the variables to 0, which would force

, so the point we found is the only extremum.
Hello :
the consecutive room numbers are : n , n+1
n(n+1) =420
n²+n -420 = 0
the discriminant delta = b² - 4ac ( a = 1 and b = 1 and c = - 420
delta = (1)²-4(1)(-420) =1681= 41²
n = (-1 -41)/2 ( refused negatif number)
or
n = (-1+41)/2 =20
<span> the room numbers.are : 20 , 21</span>
For the first part remember that an equilateral triangle is a triangle in which all three sides are equal & all three internal angles are each 60°. <span>So x-coordinate of R is in the middle of ST = (1/2)(2h-0) = h
And for the second </span><span> since this is an equilateral triangle the x coordinate of point R is equal to the coordinate of the midpoint of ST, which you figured out in the previous answer. Hope this works for you</span>
True. A triangle will always have three sides, and a pentagon will always have five.