If f is a real continuous function and f(a)f(b) < 0, then f(x) has ______ in…
2015
If f is a real continuous function and f(a)f(b) < 0, then f(x) has ______ in [a, b].
- A.
At least one root
- B.
No roots
- C.
Odd number of roots
- D.
Even number of roots
- E.
Exactly one root
Attempted by 28 students.
Show answer & explanation
Correct answer: A
Correct answer: At least one root
f(a)f(b) < 0 means f(a) and f(b) have opposite signs.
Because f is continuous on [a, b], the Intermediate Value Theorem guarantees that f(x) = 0 for at least one x in (a, b).
The exact number of roots is not determined. It may be one or more, so options such as odd number, even number, or exactly one root are not guaranteed.