16.6: Binary Searches
a tree in which every node has degree \(1\) or \(3\text{,}\) except for a single node of degree \(2\)
the unique node of degree \(2\) in a binary search tree
a node of degree \(1\) in a binary search tree
the construction of a binary search tree through a series of “either-or” decisions
Estimate the root of \(f(x) = 4x^3 + 6x^2 + 3x - 1\) that lies in \((0,1)\) to \(2\) decimal places.
Solution
The Intermediate Value Theorem from first-year calculus says that if \(f\) is continuous on the closed interval \([a,b]\) and \(f(a),f(b)\) are nonzero and opposite signs, then \(f\) has a root in the open interval \((a,b)\text{.}\) We have \(f(0) = -1 \lt 0\) and \(f(1) = 12 \gt 0\text{,}\) so there is indeed a root in \((0,1)\text{.}\) The graph in Figure \(\PageIndex{1}\) was obtained by performing a binary search by splitting into subintervals.
Since \(f(0.225) \gt 0\text{,}\) the root must be in the subinterval \((0.22,0.225)\text{.}\) This tells us to round down to \(0.22\) instead of rounding up to \(0.23\text{,}\) so we conclude that the root is approximately \(0.22\text{.}\)