Let g(x)=tan(x). Can we use the intermediate value theorem to say the equation g(x)=0 has a solution where (pi)/(4)

Question

Let  g(x)=tan(x). Can we use the intermediate value theorem to say the equation  g(x)=0 has a solution where  (pi)/(4) <= x <= (3pi)/(4) ? Choose 1 answer: A No, since the function is not continuous on that interval. (B) No, since 0 is not between  g((pi)/(4)) and  g((3pi)/(4)). (C) Yes, both conditions for using the intermediate value theorem have been met.

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Intermediate Value Theorem Application: The Intermediate Value Theorem states that if a function f is continuous on a closed interval [a, b] and k is any number between f(a) and f(b), then there exists at least one number c in the interval [a, b] such that f(c) = k. To apply this theorem to the function g(x) = tan(x) and the interval [(pi)/(4), (3pi)/(4)], we need to check two conditions: (1) that g(x) is continuous on the interval, and (2) that 0 is between g((pi)/(4)) and g((3pi)/(4)).Checking Continuity: First, we check the continuity of g(x) = tan(x) on the interval [(pi)/(4), (3pi)/(4)]. The tangent function has discontinuities at odd multiples of (pi)/2, because it is undefined there (the cosine function in the denominator of tan(x) = sin(x)/cos(x) is zero at these points). Since (pi)/2 is within the interval [(pi)/(4), (3pi)/(4)], the function g(x) is not continuous on the entire interval.Conclusion: Since g(x) = tan(x) is not continuous on the interval [(pi)/(4), (3pi)/(4)], we cannot use the Intermediate Value Theorem to guarantee that there is a solution to g(x) = 0 on this interval. Therefore, the correct answer is A: No, since the function is not continuous on that interval.

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