Posted by Soroban on September 22, 2002 at 10:38:30:
In Reply to: Polynomial Function posted by Nabill on September 22, 2002 at 00:35:23:
: A polynomial function f(x) of degree 4 has roots at x = -3, x = 0, x = 2, and x = 4. The function g(x) = f(x) + A has only one root, at x = k. What is the minimum value of f(x)? (A>0)
What an interesting problem! It has a "Think about it" solution.
f(x) is a quartic function with four x-intercepts, and opens upward.*
g(x) = f(x) + A: this is the same curve, moved upward A units. (A > 0)
And now it has only one x-intercept (k,0), which must be the minimum of g(x).
Move it back down A units: f(x) has a minimum at x = k: f(k) = -A.
* It is now obvious that the quartic curve f(x) opens upward.
Consider: if it opened downward and had four x-intercepts, then when
it is raised A units, it would still have at least two x-intercepts.
Post a Followup