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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Algebra 2

Complex conjugate theorem

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Q. 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.)

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 $f'(c) = \frac{f(b) - f(a)}{b - a}$ . 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 $\frac{1}{x}$ , and the derivative of $x$ is $1$ . So, $f'(x) = 9 \ln(x) + 9 \cdot (\frac{1}{x}) \cdot x = 9 \ln(x) + 9$ .
Evaluate Function Values: Evaluate $f(1)$ and $f(9)$ to find the average rate of change over the interval $[1, 9]$ . Since $\ln(1) = 0$ , $f(1) = 9 \times 1 \times 0 + 9 = 9$ . To find $f(9)$ , we calculate $9 \times 9 \times \ln(9) + 9$ .
Calculate Average Rate: Calculate $f(9) = 9 \times 9 \times \ln(9) + 9$ . We know that $\ln(9)$ is approximately $2.197$ (using a calculator), so $f(9) \approx 9 \times 9 \times 2.197 + 9 = 178.761 + 9 = 187.761$ .
Set Derivative Equal: Now, calculate the average rate of change $(f(b) - f(a)) / (b - a)$ using $f(1) = 9$ and $f(9) \approx 187.761$ . 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 $\ln(c)$ : $\ln(c) = \frac{13.345125}{9}$ . This simplifies to $\ln(c) \approx 1.482791667$ .
Calculate Approximate Value: Exponentiate both sides to solve for $c$ : $e^{\ln(c)} = e^{1.482791667}$ . This simplifies to $c \approx e^{1.482791667}$ .
Calculate Approximate Value: Exponentiate both sides to solve for $c$ : $e^{\ln(c)} = e^{1.482791667}$ . This simplifies to $c \approx e^{1.482791667}$ .Use a calculator to find the value of $c$ : $c \approx e^{1.482791667} \approx 4.405$ .