Which theorem can be used to determine whether a function f(x) has any zeros in a given interval?

Solution:

Intermediate value theorem can be used which states that, if a function f(x) has a zero in [a, b] and f(a) and f(b) must be of opposite signs then there exists at least one c belongs to [a, b] such that f(c) = 0.

Example: Consider f(x) = (1 - x)(x + 2) in [-3, 0]

Here f(a) = f(-3) = (1 - (-3)) (-3 + 2) = 4 × (-1) = -4

f(b) = f(0) = (1 - 0)(0 + 2) = 1 × 2 = 2

We observe that f(a) and f(b) are of opposite signs. There exists x = -2 such that f(-2) = (1- (-2))(-2+2) = 0 and -2 belong to [-3, 0]. This verifies the Intermediate value theorem.

Which theorem can be used to determine whether a function f(x) has any zeros in a given interval?

Summary:

Intermediate value theorem can be used to determine whether a function f(x) has any zeros in a given interval.