Let g(x)=cos(x). Can we use the intermediate value theorem to say the equation g(x)=0.8 has a solution where 0

Question

Let  g(x)=cos(x). Can we use the intermediate value theorem to say the equation  g(x)=0.8 has a solution where  0 <= x <= (pi)/(2)? Choose 1 answer: A No, since the function is not continuous on that interval. (B) No, since 0.8 is not between  g(0) and  g((pi)/(2)). (c) Yes, both conditions for using the intermediate value theorem have been met.

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Definition of Intermediate Value Theorem: 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. We need to check if g(x) = cos(x) is continuous on the interval [0, π/2] and if 0.8 is between g(0) and g(π/2).Check Continuity of g(x): First, we check the continuity of g(x) = cos(x) on the interval [0, π/2]. The cosine function is continuous everywhere on the real number line, including the interval [0, π/2].Evaluate g(0) and g(π/2): Next, we evaluate g(0) and g(π/2). We have g(0) = cos(0) = 1 and g(π/2) = cos(π/2) = 0. We need to determine if 0.8 is between these two values.Determine Position of 0.8: Since 0.8 is between 0 and 1, it is between g(0) and g(π/2). Therefore, the value 0.8 is indeed between the values of g(x) at the endpoints of the interval [0, π/2].Conclusion of Intermediate Value Theorem: Since both conditions for the intermediate value theorem are met (the function is continuous on the interval and the value 0.8 is between g(0) and g(π/2)), we can conclude that there is at least one solution to the equation g(x) = 0.8 in the interval [0, π/2].

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