Find all points c satisfying the conclusion of the MVT for the function f(x)=9x ln(x)+9 and interval [1,9]. (Use decimal notation. Give your answer to three decimal places. Give your answer as comma-separated list.)

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Find all points  c satisfying the conclusion of the MVT for the function  f(x)=9x ln(x)+9 and interval  [1,9]. (Use decimal notation. Give your answer to three decimal places. Give your answer as comma-separated list.)

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Mean Value Theorem: The Mean Value Theorem states that if a function f 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 (a, b) such that f'(c) = (f(b) - f(a)) / (b - a). We need to find the derivative of f(x), which is f'(x).Calculate Derivative: Calculate the derivative of f(x) = 9x ln(x) + 9 using the product rule and the chain rule. The derivative of ln(x) is 1/x, and the derivative of x is 1. So, f'(x) = 9 ln(x) + 9 * (1/x) * x = 9 ln(x) + 9.Evaluate Function Values: Evaluate f(1) and f(9) to find the average rate of change over the interval [1, 9]. Since ln(1) = 0, f(1) = 9 * 1 * 0 + 9 = 9. To find f(9), we calculate 9 * 9 * ln(9) + 9.Calculate Average Rate: Calculate f(9) = 9 * 9 * ln(9) + 9. We know that ln(9) is approximately 2.197 (using a calculator), so f(9) ≈ 9 * 9 * 2.197 + 9 = 178.761 + 9 = 187.761.Set Derivative Equal: Now, calculate the average rate of change (f(b) - f(a)) / (b - a) using f(1) = 9 and f(9) ≈ 187.761. The average rate of change is (187.761 - 9) / (9 - 1) = 178.761 / 8 = 22.345125.Isolate Natural Logarithm: Set the derivative f'(c) equal to the average rate of change to find c. So, we have 9 ln(c) + 9 = 22.345125. We need to solve this equation for c.Solve for ln(c): Subtract 9 from both sides of the equation to isolate the natural logarithm term: 9 ln(c) = 22.345125 - 9. This simplifies to 9 ln(c) = 13.345125.Exponentiate to Find c: Divide both sides of the equation by 9 to solve for ln(c): ln(c) = 13.345125 / 9. This simplifies to ln(c) ≈ 1.482791667.Calculate Approximate Value: Exponentiate both sides to solve for c: e^(ln(c)) = e^(1.482791667). This simplifies to c ≈ e^(1.482791667).Use a calculator to find the value of c: c ≈ e^(1.482791667) ≈ 4.405.

Find all points $c$ satisfying the conclusion of the MVT for the function $f(x)=9 x \ln (x)+9$ and interval $[1,9]$ . (Use decimal notation. Give your answer to three decimal places. Give your answer as comma-separated list.)

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Find all points c c c satisfying the conclusion of the MVT for the function f(x)=9xln⁡(x)+9 f(x)=9 x \ln (x)+9 f(x)=9xln(x)+9 and interval [1,9] [1,9] [1,9]. (Use decimal notation. Give your answer to three decimal places. Give your answer as comma-separated list.)

Find all points $c$ satisfying the conclusion of the MVT for the function $f(x)=9 x \ln (x)+9$ and interval $[1,9]$ . (Use decimal notation. Give your answer to three decimal places. Give your answer as comma-separated list.)