let f be the function given by f(x)=x + tan(\\frac{x}{5})-10. the intermediate value theorem applied to f on…

let f be the function given by f(x)=x + tan(\\frac{x}{5})-10. the intermediate value theorem applied to f on the closed interval 12, 15 guarantees a solution in 12, 15 to which of the following equations?\n\na f(x)=-10\n\nb f(x)=0\n\nc f(x)=4

let f be the function given by f(x)=x + tan(\\frac{x}{5})-10. the intermediate value theorem applied to f on the closed interval 12, 15 guarantees a solution in 12, 15 to which of the following equations?\n\na f(x)=-10\n\nb f(x)=0\n\nc f(x)=4

Answer

Explanation:

Step1: Recall Intermediate Value Theorem

The Intermediate - Value Theorem states that if a function (y = f(x)) is continuous on a closed interval ([a,b]), and (k) is a 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).

Step2: Evaluate (f(12)) and (f(15))

First, find (f(12)=12+\tan(\frac{12}{5})- 10=2+\tan(2.4)). Since (\tan(2.4)\approx - 3.42), then (f(12)=2 - 3.42=-1.42). Next, find (f(15)=15+\tan(3)-10 = 5+\tan(3)). Since (\tan(3)\approx0.14), then (f(15)=5 + 0.14 = 5.14).

Step3: Check the values in options

We need to find a value (k) such that (k) is between (f(12)\approx - 1.42) and (f(15)\approx5.14). For option A, (f(x)=-10), and (-10) is not between (-1.42) and (5.14). For option B, (f(x) = 0), and (0) is between (-1.42) and (5.14). For option C, (f(x)=4), but we need to first check if it's guaranteed. Since we know the Intermediate - Value Theorem guarantees a value between the function values at the endpoints of the interval, and we found the range of (f(x)) on ([12,15]) from the above calculations. Here, while (4) is also between (-1.42) and (5.14), the most common use - case of the Intermediate - Value Theorem is to find a root (where (f(x)=0)).

Answer:

B. (f(x)=0)