x + y + z = 360 and sinx*siny*sinz
-> f(x,y) = sin(x)*sin(y)*sin(360-x-y)
-> f(x,y) = sin(x)*sin(y)*-sin(x+y)
Now to find the critical points of f(x,y), we just need to find the partial derivative with respect to x and y and solve for 0.
f_x(x,y) =
sin(x)sin(y)(-cos(x+y)-cos(x)sin(y)sin(x+y)
solve for 0.
sin(x)sin(y)(-cos(x+y)-cos(x)sin(y)sin(x+y)=0
-> tan(x) = -tan(x+y)
Since the function is symmetric, we should get the same partial derivative for y.
-> tan(y) = -tan(y+x)
-> tan(x)=tan(y)
-> x = y or they are opposites. However, if they are opposites, the original function just becomes 0. Thus, x = y.
Now substitute in x for y in the original equation and find its critical points.
Eventually, you will get sin(3x)=0
x = 120 degrees
y = 120 degrees
z = 120 degrees
Answer (D)
Dervish Fox: I see some beauty between this one and Fox 316.
Red Fox: Would you please help me see that beauty?
Dervish Fox: The term "sin(B)+sin(C)-sin(A)" has evolved into "sin(B+C-A)" in this one. And the rest is identical.
Red Fox: And you found that beautiful?
Dervish Fox: Both answers are very different and very similar at the same time. Don't you feel anything about that?
Red Fox: I am a rational guy. I depend on my intelligence only. I can easily prove both and there is nothing magical or irrational about that. Everything's explainable. Pure and simple!
Dervish Fox: And is that all you can be? Is that the only direction you can grow? Is that who you are, or are you more than that? Why walk when you can fly? Why mutter when you can sing? Why scribble when you can paint?
Red Fox: I think I am starting to feel little bit of annoyance.
Dervish Fox: Not bad for a start my friend, not bad at all !
This means that the point M we are looking for is such that M is midpoint of AB and AB tangent in M. This brings us to problem 260 where we already encountered point V(4/3, 8/9):
- tangent in V is parallel to UP and vector(UP) = vector(VB) because of parabola property (see Fox 260)
- vector(UP) is symmetric of vector(VO) along y axis because of parabola symmetry
- thus OVB is isoceles and V is the midpoint of AB
Thus V achieves minimal OAB area.
Area = 4/3 . 16/9 = 64/27.
By Yu:
Equation of tangent at x=a, y=-(2a-2)x+a^2.
Area of triangle = (a^4)/(4(a-1)).
When a=4/3, minimum area = 64/27.
Question: a=4/3 in Fox 260 and Fox 262. Is this a coincidence?
Then let's assume U and V achieve the maximum trapezoid area between A (x=0) and B(x=2), then necessarily Au=uv and uv=vB which implies Au=uv=vB=AB/3=2/3. Because of symmetry, maximum area is equivalent to area of rectangle AvVW = 8/9 * 4/3 = 32/27.
Geometric Translation
by Yu
Translate y=2x-x^2 to the left by 1 unit to obtain y=1-x^2.
Without going into details, the area of the trapezium is greater than the area of the quadrilateral. Area of trapezium, A = (1/2) (2x+2)(1-x^2) = (x+1)(1-x^2)
Max A = 32/27 when x = 1/3.
For more details see Fox 260.
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