How many roots does the following quadratic function of t have: f(t) = a + bt + ct^2 if it is known that f(t) >=0 for all values of t ?

To write more clearly, the function is;
#f(t) = a + bt + ct^2#
The function f has:
a) Exactly 2 roots
b) no roots
c) Exactly one root
d) At most one root

2 Answers
Dec 21, 2017

d) At most one root.

Explanation:

There are two possible conditions that satisfy the specification:

#f(t)>= 0#

The above says that all of the points have a y value greater than 0, except, perhaps, 1 point has a y value that equals 0. The inequality does not guarantee that 1 point has a y value that is equal to 0 but it leaves it as a possibility. Therefore, the correct selection is:

d) At most one root.

Dec 21, 2017

At most one root

Explanation:

The graph of #f# is a parabola that either opens upward or downward.
Since #f(t) >= 0# for all #t#, the parabola must open upward.

An upward opening parabola can intersect the #x# axis in #0#, #1#, or #2# points, but if it intersects in 2 points, the the values of #f(t)# must be negative between the intercepts.

Therefore, the parabola intersects the #x# axis in #1# or #0# points, and the function has at most one root.