Use Genenalized mean value twoorem to estalslish the following inequality for x 50 : (1)/(2(x+1)e^(2x))

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Use Genenalized mean value twoorem to estalslish the following inequality for  x 50 :  (1)/(2(x+1)e^(2x)) < (ln(x+1))/(e^(2x)-1) < (1)/(2)

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Introduction: Let's consider two functions f(x) and g(x) that are continuous on the closed interval [a, b] and differentiable on the open interval (a, b). The Generalized Mean Value Theorem (also known as Cauchy's Mean Value Theorem) states that there exists some c in (a, b) such that: (f(b) - f(a)) / (g(b) - g(a)) = f'(c) / g'(c) For our problem, we need to choose appropriate f(x) and g(x) to apply the theorem. Let's take f(x) = ln(x+1) and g(x) = e^(2x) - 1. We will apply the theorem on the interval [0, x] for x > 0.Verification of Conditions: First, we need to verify that f(x) and g(x) satisfy the conditions of the theorem. The function f(x) = ln(x+1) is continuous and differentiable for x > 0, and g(x) = e^(2x) - 1 is also continuous and differentiable for all x since the exponential function is continuous and differentiable everywhere. Thus, the conditions are satisfied.Derivative Calculation: Now, let's calculate the derivatives f'(x) and g'(x). For f(x) = ln(x+1), the derivative is f'(x) = 1/(x+1). For g(x) = e^(2x) - 1, the derivative is g'(x) = 2e^(2x).Applying Generalized Mean Value Theorem: Applying the Generalized Mean Value Theorem, we get: (ln(x+1) - ln(1)) / (e^(2x) - e^(0)) = (1/(c+1)) / (2e^(2c)) Simplifying the left side, we have: ln(x+1) / (e^(2x) - 1) = (1/(c+1)) / (2e^(2c)) Now, we need to find the bounds for the right side of the equation.Finding Bounds: Since c is in the interval (0, x), we know that c > 0. Therefore, c+1 > 1 and e^(2c) > e^(0) = 1. This means that 1/(c+1) < 1 and 2e^(2c) > 2. So, the fraction (1/(c+1)) / (2e^(2c)) is less than 1/2.On the other hand, as c approaches 0 from the right, 1/(c+1) approaches 1 and 2e^(2c) approaches 2, so the fraction (1/(c+1)) / (2e^(2c)) approaches 1/2 from the left. Therefore, we have: (1)/(2(x+1)e^(2x)) < (1/(c+1)) / (2e^(2c)) < (1)/(2) Since ln(x+1) / (e^(2x) - 1) = (1/(c+1)) / (2e^(2c)), we can conclude that: (1)/(2(x+1)e^(2x)) < ln(x+1) / (e^(2x) - 1) < (1)/(2)

Use Genenalized mean value twoorem to estalslish the following inequality for $x 50$ : $\newline$ \frac{1}{2(x+1) e^{2 x}}<\frac{\ln (x+1)}{e^{2 x}-1}<\frac{1}{2}

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Use Genenalized mean value twoorem to estalslish the following inequality for x50 x 50 x50 :\newline \frac{1}{2(x+1) e^{2 x}}<\frac{\ln (x+1)}{e^{2 x}-1}<\frac{1}{2}

Use Genenalized mean value twoorem to estalslish the following inequality for $x 50$ : $\newline$ \frac{1}{2(x+1) e^{2 x}}<\frac{\ln (x+1)}{e^{2 x}-1}<\frac{1}{2}