Known function $ f (x) = - x ^ 3 + 9x ^ 2-26x + 27 $, for any $ k> 0 $, straight $ y = kx + a $ curve $ y = f (x) $ has the only common point, find the range of $ a $.
Analysis: $ f ^ { "} ( x) = - 6x + 18 $, as shown, $ f (x) $ in $ (- \ infty, 3) $ downwardly convex, $ (3, + \ infty ) $ on projections. inflection $ P (3, f (3 )) is tangent at $ equation Y = X $ $.
$ F (X) of the maximum value of $ $ M (3+ \ dfrac {\ sqrt {3}} {3}, 3 + \ dfrac {2 \ sqrt {3}} {9}) $, denoted $ Q (0, a) $ apparent from FIG.
1) $ a \ ge3 + \ dfrac {2 \ sqrt {3}} {9} $ when $ y = kx + a $ and $ y = f (x) $ image there is only one common point.
2) $ a \ in [3,3+ \ dfrac {2 \ sqrt {3}} {9}) $ , since $ k> 0 $ therefore, three common points.
. 3) $ A \ in (0,3 ) $, the three common points.
. 4) $ A \ in (- when \ infty, 0] $,
if $ k \ in (0, k_ {PQ}) convex portion $, lines and curves has a unique common point ;
if $ k = k_ {PQ} $ lines and curves has a unique common point $ P $;
. if $ k \ in (k_ {PQ }, + \ infty) the convex portion of the lower $, lines and curves with a single common point
heald on, $ a \ in (- \ infty, 0] \ cup [3+ \ dfrac {2 \ sqrt {3}} {9}, + \ infty) $