If f(1) > 0 and f(3) < 0, then there exists a number c between 1 and 3 such that f(c) = 0. Is this true or false?
Please and thank you
Absolutely true, uuuuhhh, if f(x) is continuous.
Can you please explain why it is true
Please and Thank you
If the curve is continuous, and at x=1 f(x) is positive, and at x=3, f(x) is negative, somewhere it crossed the axis.
To determine whether the statement is true or false, we can use the Intermediate Value Theorem (IVT). The IVT states that if a function f is continuous on the closed interval [a, b] and f(a) is less than f(b), then for any value y between f(a) and f(b), there exists a number c in the interval [a, b] such that f(c) = y.
In this case, since f(1) > 0 and f(3) < 0, we have f(1) > f(3), satisfying the condition required by the IVT. Therefore, according to the Intermediate Value Theorem, there must exist a number c between 1 and 3 such that f(c) = 0. Hence, the statement is true.