The function f(x) is continuous on the interval [1,10]. The table below gives some of its values. What is the minimum number of zeros that f(x) is guaranteed to have by the Intermediate Value Theorem?a graph shows x values 1-10 and f(x) values -4, 4, 1, 2, -5, -3, 8, 6, 8, -4
To determine the minimum number of zeros that the function f(x) is guaranteed to have by the Intermediate Value Theorem, we need to examine the behavior of f(x) between the given values
To determine the minimum number of zeros that the function f(x) is guaranteed to have by the Intermediate Value Theorem, we need to examine the behavior of f(x) between the given values.
The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on values of opposite signs at the endpoints f(a) and f(b), then there exists at least one number c in the interval (a, b) such that f(c) = 0.
Looking at the given table, we can observe that f(1) = -4 and f(10) = -4. These values have the same sign, which means they do not satisfy the condition required by the Intermediate Value Theorem for having a zero. Therefore, we can eliminate the numbers 1 and 10 from consideration as potential zeros of f(x).
Next, we look for any changes in sign between the remaining values. From the table, we notice that f(5) = -5 and f(6) = -3. These two values have opposite signs, indicating that there is a sign change between them. According to the Intermediate Value Theorem, there must exist at least one zero of f(x) between x = 5 and x = 6.
Therefore, the minimum number of zeros that f(x) is guaranteed to have by the Intermediate Value Theorem is one.
It’s important to note that there could be additional zeros between the given values that are not apparent from the table. However, based solely on the given information, we can guarantee the existence of at least one zero between x = 5 and x = 6.
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