be the unique positive real number satisfying ![\[ 2^x+32^x=8^x+16^x. \]](http://latex.artofproblemsolving.com/2/5/8/258d9eaddb9352a4f72951eff657d8559553d722.png)
Compute 
.
increasing with the limit to 
is 
converge ?
is a complex matrix with 
prove that 
be a continuously differentiable function, and let 
be real numbers such that 
.
such that
be a function such that 
for real numbers 
and 
Prove that there exists a function 
such that 
for all real numbers 
and 
)
or
be a vector space over 
and 
be a linear transformation such that 
and 
are subspaces of 
.
along 
*
defined as follows: for each 
.
is a projection
.
.
irrational number or rational number？ Any comments or hints will welcome!
such that 
is rational ?
Y by
Consider this parabola: y = x^2 + (2m + 1)x + m(m - 3) where m is constant and -1 ≤ m ≤ 4. A(-m-1, y1), B(m/2, y2), C(-m, y3) are three different points on the parabola. Now rotate the axis of symmetry of the parabola 90 degrees counterclockwise around the origin O to obtain line a. Draw a line from the vertex P of the parabola perpendicular to line a, meeting at point H.
1) express the vertex of the quadratic equation using an expression with m.
2) If, regardless of the value of m, the parabola and the line y=x−km (where k is a constant) have exactly one point of intersection, find the value of k.
3) (where I'm struggling the most) When 1 < PH ≤ 6, compare the values of y1, y2, and y3.
1) express the vertex of the quadratic equation using an expression with m.
2) If, regardless of the value of m, the parabola and the line y=x−km (where k is a constant) have exactly one point of intersection, find the value of k.
3) (where I'm struggling the most) When 1 < PH ≤ 6, compare the values of y1, y2, and y3.
;
.
in exactly one point, then:![$x^2 + (2m+1)x + m(m-3) = x-km \Rightarrow x^2 +2mx + (m^2-3m+km) = 0 \Rightarrow x = \frac{-2m \pm \sqrt{4m^2-4(1)[m^2+(k-3)m]}}{2}$](http://latex.artofproblemsolving.com/5/3/5/5355e159ad0ee4c83a120507190a277bae3d8690.png)
(i);
.![$m \in [-1,4]$](http://latex.artofproblemsolving.com/5/9/a/59a272a2cbe75fb8ca460a80c5b4b88963852016.png)
we have the points 
on the parabola. Checking 
gives us the orderings:
for ![$m \in \left[-1,-\frac{2}{3}\right) \cup (0,4]$](http://latex.artofproblemsolving.com/9/2/7/927ee826fa7ba6b345ce3bd08e232088663c37fc.png)
;
for 
;
for 
.
)
counterclockwise about the origin gives us the horizontal line 
from which 
.
,![$1 < \frac{|1-12m|}{4}\le 6 \Rightarrow m \in \left[-\frac{23}{12},-\frac{1}{4}\right) \cup \left(\frac{5}{12},\frac{25}{12}\right]$](http://latex.artofproblemsolving.com/b/e/3/be3c2c6855ad0abba11fddf49f47508d0fa00c82.png)
.
![$\textcolor{red}{m \in \left[-1, -\frac{2}{3}\right) \cup \left(\frac{5}{12},\frac{25}{12}\right]}$](http://latex.artofproblemsolving.com/8/2/1/8215c0bf1f6da94684c8b8a990cb099ac9119a70.png)
;
;
.
