Information about concavity is contained in the second derivative of a function. Given f(x) = ax² + bx + c, we have
f'(x) = 2ax + b
and
f''(x) = 2a
Concavity changes at a function's inflection points, which can occur wherever the second derivative is zero or undefined. In this case, since a ≠ 0, the function's concavity is uniform over its entire domain.
(i) f is concave up when f'' > 0, which occurs when a > 0.
(ii) f is concave down when f'' < 0, and this is the case if a < 0.
In Mathematica, define f by entering
f[x_] := a*x^2 + b*x + c
Then solve for intervals over which the second derivative is positive or negative, respectively, using
Reduce[f''[x] > 0, x]
Reduce[f''[x] < 0, x]
Its a rectangle so two sides are equal and both sides that are opposite of each other are equal. So two sides are 37. Then you add them together to get 74, then subtract it from 164 to get 90 the rest of the fence length. Then divide it by two to get 45 which is the other side lengths, so it would be a 37 by 45 rectangle one side would be 37 while the other two are 45.
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AD = DB
Angle ADE = Angle CDB
Angle DAE = Angle DBC (Alternate angles)
Angle DEA = Angle DCB (Alternate angles)
Since you have two congruent angles and one congruent side, triangle ADE is congruent to triangle CDB. This means that DE is congruent to DC implying D is the midpoint of CE.
