Rom

latex interpreter fails

If we restrict the domain from the vertex to positive infinity then f(x) will be a bijection.

We can read the vertex off as x=-2, so we restict the domain as -2 <= x < +infinity

1) do the braces mean ceiling or fractional part or neither?

2) clearly x = z + 0.8

a little tinkering shows x = 1.8

2 + 1 + 1.8 = 4.8

11 possible each pick, 4 picks, total of (11)⁴ = 14641 arrangements

L = A  + (B-A)t = (1+t, 1+2t, 2+2t)

a=1, b=1, c=2, d=1, e=2, f=2

(c/a, e/a) = (2,2)

\(\text{Just by looking at it I maintain the answer is 3, but we'll do it formally}\\ \nabla\left(\dfrac a b + \dfrac b c + \dfrac c a\right) = 0\\ \dfrac 1 b - \dfrac{c}{a^2} = 0\\ \dfrac 1 c - \dfrac{a}{b^2} = 0\\ \dfrac 1 a - \dfrac{b}{c^2} = 0\)

\(\text{The only solution of all positive numbers is $a=b=c$}\\ \text{and the extrema value is 3}\\ \text{Suppose $c=2a$}\\ \dfrac a b + \dfrac b c + \dfrac c a = 1+\dfrac 1 2 + 2 = \dfrac 7 2 > 3\\ \text{we know 3 is either a min or a max and thus as $3 < \dfrac 7 2$ it must be a min}\)

\(\sqrt{x y} = \sqrt{3}\\ \dfrac{2}{\dfrac 1 x + \dfrac 1 y}=\dfrac 3 2\\ \dfrac{2xy}{x+y}=\dfrac 3 2\\ \dfrac{6}{x+y}=\dfrac 3 2\\ x+y = 4\\ xy=3\\ x+\dfrac 3 x = 4\\ x^2-4x+3=0\\ (x-3)(x-1)=0\\ x=3,1\\ (x,y) = (1,3), (3,1)\)

the way they want you to look at this problem is to determine whether the operations listed can be reversed to

obtain the original value.

For example to reverse D we find a matrix that sends every vector to 1/2 of itself.

See if you can apply this thinking to the others.

1)

\(\text{well we can see without much difficulty that $x \in (8, 9)$ somewhere}\\ \text{in that range $\lfloor x \rfloor = 8$}\\ \text{So it must be that}\\ x^2 = 75-8 = 67\\ x = \sqrt{67}\)

2)

\(|y| - \dfrac{2x}{|x|} = -1\\ |y|-2\text{sgn$(x)= -1$}\\ |y|=-1 + 2\text{sgn$(x)$}\\ \text{The only solutions to this are }\\ x>0,~y = \pm 1\)

\(\text{applying the results of the previous equation to the second}\\ x^2 +1 = 24\\ x^2 -1 = 24\\ x =\sqrt{23}, ~ 5\\ \text{the ordered pairs of solutions are }\\ (5,-1),~(\sqrt{23},1)\\ \text{and the sum the problem asks for is $\sqrt{23}+5$} \)

a few lines of mathematica shows the smallest is 2666, and the largest is 9443

9443-2666 = 6777