Final answer:
The Intermediate Value Theorem shows that sin x - cos x = 3x has a solution, while Rolle's Theorem proves that this solution is unique.
Step-by-step explanation:
The Intermediate Value Theorem states that if a function is continuous on a closed interval and takes on two different values at the endpoints of the interval, then it must also take on every value between those two values at some point within the interval.
In this case, we have the equation sin x - cos x = 3x. To show that it has a solution, we can rewrite it as sin x - cos x - 3x = 0. Let's define a new function, f(x) = sin x - cos x - 3x. Now we need to find two values of x, say a and b, such that f(a) and f(b) have opposite signs. We can choose a = 0 and b = π, where π is pi. Then, f(0) = -1 and f(π) = π - 1. Since f(a) and f(b) have opposite signs, by the Intermediate Value Theorem, there exists at least one value of x between 0 and π for which sin x - cos x = 3x has a solution.
To show that this solution is unique, we can use Rolle's Theorem, which states that if a function is continuous on a closed interval, differentiable on the open interval, and takes on the same value at the endpoints of the interval, then there exists at least one point in the interval where the derivative of the function is equal to zero.
In this case, let's define another function, g(x) = sin x - cos x - 3x. The function g(x) is the same as f(x) defined earlier, but with a different name. We already know that f(x) has a solution between 0 and π. Now, we need to show that the derivative of g(x) equals zero at this point. Taking the derivative of g(x), we have g'(x) = cos x + sin x - 3. Evaluating g'(x) at our solution point, say c, we get g'(c) = cos c + sin c - 3. Since g(a) = g(b), that means g(c) = g(0) = g(π). Therefore, g'(c) = cos c + sin c - 3 = 0. Hence, by Rolle's Theorem, the solution to sin x - cos x = 3x is unique.
Learn more about Intermediate Value Theorem, Rolle's Theorem, uniqueness of solution