Square on the curve A square is drawn on the curve $y = x^3 + 27\cdot x^2 + 8\cdot x + 91$. What would be the area of the square. All the points of the square should lie on the curve.
All four points lie on the curve (not the sides)
 A: Time for some reverse engineering.  Suppose we start with a square; translate it so that it is centred at the origin.  Then for some $s$ and $\theta$, the vertices are $(s \cos(\theta), s \sin(\theta))$, $(s \sin(\theta), -s \cos(\theta))$, $(-s \cos(\theta), -s \sin(\theta))$ and $(-s \sin(\theta), s \cos(\theta))$, and the area is $2 s^2$.  The cubic that interpolates these points is $$f(x) = \frac{4}{s^2 \sin(4 \theta)} x^3 - \frac{3 + \cos(4 \theta)}{\sin(4 \theta)} x$$
Note that for $b \ge \sqrt{8}$, $b = (3 +  \cos(4 \theta))/\sin(4 \theta)$ has  real solutions, namely $$\sin(4 \theta) = \frac{3b \pm \sqrt{b^2-8}}{b^2  - 1}$$ Note also that  these solutions have $\sin(4 \theta) > 0$.  We can then take $s=2/\sqrt{\sin(4\theta)}$ to make the cubic $f(x) = x^3 - b x$, and the area is 
$$2s^2 = \frac{8}{\sin(4\theta)} = 3 b \mp \sqrt{b^2 - 8}$$  The given problem has $b = 235$  and area $705 \mp \sqrt{55217}$.
A: If you take D. Thomine's hint,  and let $X=x+9$ and $Y=y-1477$ to centre the cubic at $(0,0)$ as $Y=X^3-235X$.  So if $(X_0,Y_0)$ are vertices of the square centred on the origin then so too are $(Y_0,-X_0)$,  $(-X_0,-Y_0)$ $(-Y_0,X_0)$. 
So you need to solve $(X^3-235X)^3-235(X^3-235X)=-X$.  There are nine solutions including $X=0$; the others come in two sets of four for the two possible squares.  
One square has a corner near $(0.065234579,-15.32984851)$ and just rotate by one, two or three right angles to find the others.  The area is just over $470$. 
The other square has a corner near $(15.29691915,-15.36215329)$ and again rotate to find the others.  The area is just under $940$. 
To find the corners of the original squares on the original cubics, just add $(-9, 1477)$ to the coordinates of the four corners of the squares, but you do not ned to do this to find the areas.
A: Note: This answers an earlier version of the question.
It is impossible to draw a square of positive area in $\mathbb R^2$ all of whose points lie on the graph of $y=x^3+27x^2+8x+91$. Proof: Any side of the square will have to lie on that graph, so that an interval will have to lie on the graph, but an interval has zero second derivative whereas $\frac{d^2}{dx^2}x^3+27x^2+8x+91=6x+54$ which is not zero on an interval.
Thus, if you're seeking the area of a square drawn so its sides are all on that curve, the answer is $0$.
A: Suppose the points $V_i = (x_i, f(x_i))$, $i=1,2,3,4$, form the vertices of a square.  Since there is only one interval in which $f' < 0$, if $x_1 < x_2 < x_3 < x_4$ the vertices in counterclockwise order will be $V_1, V_3, V_4, V_2$.  For these to form a square we need
$V_4 - V_2 = V_3 - V_1$, $\|V_3 - V_1\|^2 = \|V_2 - V_1\|^2$, and $(V_3 - V_1) \cdot (V_2 - V_1) = 0$.  Solving this system of four equations in four unknowns using Maple, besides the trivial solution
$x_1=x_2=x_3=x_4$ I get the following real solutions:
$$[-24.36215898, 1461.700388], [-24.29691914, 1492.362157],[6.29691914, 1461.637842], [6.362158975, 1492.299610],\  \text{ area } 705+ \sqrt{55217}$$ 
$$[-24.32984851, 1476.934762], [-9.065234, 1492.329713], [-8.934766, 1461.670288], [6.32984851, 1477.065233],\, \text{ area } 705- \sqrt{55217}$$
The decimal numbers are approximations to complicated expressions involving radicals, but the areas are exact.  Offhand I don't see why the area should come out so neatly.
