Question

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

Answer 1

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.

Answer 2

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.