Let g(x)=sqrt(5x-1) and let c be the number that satisfies the Mean Value Theorem for g on the interval [1,10]. What is c ? Choose 1 answer: (A) 2.25 (B) 4.25 (C) 6.5 (D) 8

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Let  g(x)=sqrt(5x-1) and let  c be the number that satisfies the Mean Value Theorem for  g on the interval  [1,10]. What is  c ? Choose 1 answer: (A) 2.25 (B) 4.25 (C) 6.5 (D) 8

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Define Mean Value Theorem: The Mean Value Theorem states that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the interval (a, b) such that g'(c) = (g(b) - g(a)) / (b - a). Let's calculate g'(x) first.Calculate g'(x): To find g'(x), we need to differentiate g(x) = sqrt(5x - 1). Using the chain rule, g'(x) = (1/2) * (5x - 1)^(-1/2) * 5.Find Change in g: Now we simplify g'(x) to get g'(x) = 5 / (2 * sqrt(5x - 1)).Find Change in x: Next, we calculate g(10) and g(1) to find the change in g over the interval [1, 10]. g(10) = sqrt(5*10 - 1) = sqrt(49) = 7 and g(1) = sqrt(5*1 - 1) = sqrt(4) = 2.Apply Mean Value Theorem: Now we find the change in g over the interval [1, 10], which is g(10) - g(1) = 7 - 2 = 5.Set up Equation: We also find the change in x over the interval [1, 10], which is 10 - 1 = 9.Cross-Multiply: Now we can set up the equation from the Mean Value Theorem: g'(c) = (g(10) - g(1)) / (10 - 1). Substituting the values we have, we get 5 / (2 * sqrt(5c - 1)) = 5 / 9.Solve for c: We can now solve for c by cross-multiplying: 5 * 9 = 2 * sqrt(5c - 1) * 5.Eliminate Square Root: Simplifying the equation, we get 45 = 10 * sqrt(5c - 1).Calculate c: Dividing both sides by 10, we get 4.5 = sqrt(5c - 1).Squaring both sides to eliminate the square root, we get 4.5^2 = (5c - 1).Calculating 4.5^2 gives us 20.25 = 5c - 1.Adding 1 to both sides, we get 20.25 + 1 = 5c.Simplifying, we get 21.25 = 5c.Finally, dividing both sides by 5, we find c = 21.25 / 5 = 4.25.

Let $g(x)=\sqrt{5 x-1}$ and let $c$ be the number that satisfies the Mean Value Theorem for $g$ on the interval $[1,10]$ . $\newline$ What is $c$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $2$ . $25$ $\newline$ (B) $4$ . $25$ $\newline$ (C) $6$ . $5$ $\newline$ (D) $8$

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Let g(x)=5x−1 g(x)=\sqrt{5 x-1} g(x)=5x−1​ and let c c c be the number that satisfies the Mean Value Theorem for g g g on the interval [1,10] [1,10] [1,10].\newlineWhat is c c c ?\newlineChoose 111 answer:\newline(A) 222.252525\newline(B) 444.252525\newline(C) 666.555\newline(D) 888

Let $g(x)=\sqrt{5 x-1}$ and let $c$ be the number that satisfies the Mean Value Theorem for $g$ on the interval $[1,10]$ . $\newline$ What is $c$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $2$ . $25$ $\newline$ (B) $4$ . $25$ $\newline$ (C) $6$ . $5$ $\newline$ (D) $8$