Could anyone help me on this:
how to prove, that with all kind of a, b and c value, function's graph and Ox axis has at least one common point.
What should I start at? Could you lead me to the right direction? thank you.
Hello, Ellla!
. .
An -intercept occurs when
We have: .
. . . .
Quadratic Formula:
. .
. . . .
. . . .
. . . . .[1]
If , [1] has one value.
. . The function has one -intercept.
For any other values of , [1] has two values.
. . The function has two -intercepts.
One way it to see that f(a) = -c^2 <= 0. On the other hand, if x0 > max(|a|, |b|) + |c|, then f(x0) > 0. Therefore, f(x) must assume all intermediate values between f(a) and f(x0), including 0.
Strictly speaking, this proof invokes the Intermediate value theorem from calculus, but it is one of the most obvious-looking theorems (not to say that its proof is trivial). Students of calculus are often puzzled why it has to be proved at all.