The function \( f \) is defined by \( f(x)=x^{3}-5 \cos (2 x-2) \). Find all values of \( x \) that satisfy th conclusion of the Mean Value Theorem on the interval \( [-2,3] \). You may use a calculator and round to the nearest thousandth.

To apply the Mean Value Theorem (MVT), we first need to ensure that the function \( f(x) = x^3 - 5 \cos(2x - 2) \) meets the conditions of the theorem on the interval \([-2, 3]\). The conditions are: 1. \( f \) is continuous on the closed interval \([-2, 3]\). 2. \( f \) is differentiable on the open interval \((-2, 3)\). Since \( f(x) \) is a polynomial and a composition of continuous functions, it is continuous and differentiable everywhere, including on the specified interval. Next, we need to find the average rate of change of \( f \) over the interval \([-2, 3]\): \[ \text{Average Rate of Change} = \frac{f(3) - f(-2)}{3 - (-2)} \] Let's calculate \( f(3) \) and \( f(-2) \): 1. Calculate \( f(3) \): \[ f(3) = 3^3 - 5 \cos(2 \cdot 3 - 2) = 27 - 5 \cos(4) \] 2. Calculate \( f(-2) \): \[ f(-2) = (-2)^3 - 5 \cos(2 \cdot (-2) - 2) = -8 - 5 \cos(-6) \] Now, we will compute these values. Calculate the value by following steps: - step0: Calculate: \(27-5\cos\left(4\right)\) - step1: Calculate the trigonometric value: \(30.268218\) Calculate or simplify the expression \( -8 - 5 * \cos(-6) \). Calculate the value by following steps: - step0: Calculate: \(-8-5\cos\left(-6\right)\) - step1: Calculate: \(-8-5\cos\left(6\right)\) - step2: Calculate the trigonometric value: \(-12.800851\) Now we have the values: - \( f(3) \approx 30.268 \) - \( f(-2) \approx -12.801 \) Next, we can calculate the average rate of change: \[ \text{Average Rate of Change} = \frac{f(3) - f(-2)}{3 - (-2)} = \frac{30.268 - (-12.801)}{5} = \frac{30.268 + 12.801}{5} \] Let's compute this value. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(30.268+12.801\right)}{5}\) - step1: Remove the parentheses: \(\frac{30.268+12.801}{5}\) - step2: Add the numbers: \(\frac{43.069}{5}\) - step3: Convert the expressions: \(\frac{\frac{43069}{1000}}{5}\) - step4: Multiply by the reciprocal: \(\frac{43069}{1000}\times \frac{1}{5}\) - step5: Multiply the fractions: \(\frac{43069}{1000\times 5}\) - step6: Multiply: \(\frac{43069}{5000}\) It seems that the equation cannot be solved directly using the tool. Instead, we can approach this problem numerically or graphically to find the values of \( c \) that satisfy the equation \( 3c^2 + 10 \sin(2c - 2) = 8.614 \). To do this, we can rearrange the equation to: \[ 3c^2 + 10 \sin(2c - 2) - 8.614 = 0 \] We can use numerical methods or a graphing approach to find the roots of this equation in the interval \((-2, 3)\). Let's evaluate \( f'(c) \) at several points in the interval to find where it is approximately equal to \( 8.614 \). I will check a few values of \( c \) in the interval \((-2, 3)\). Calculate the value by following steps: - step0: Calculate: \(3\times 1^{2}+10\sin\left(2\times 1-2\right)\) - step1: Multiply: \(3\times 1^{2}+10\sin\left(2-2\right)\) - step2: Subtract the terms: \(3\times 1^{2}+10\sin\left(0\right)\) - step3: Evaluate the power: \(3\times 1+10\sin\left(0\right)\) - step4: Calculate: \(3\times 1+10\times 0\) - step5: Multiply the numbers: \(3\times 1+0\) - step6: Multiply: \(3+0\) - step7: Remove 0: \(3\) Calculate or simplify the expression \( 3*(0)^{2} + 10*\sin(2*(0) - 2) \). Calculate the value by following steps: - step0: Calculate: \(3\times 0^{2}+10\sin\left(2\times 0-2\right)\) - step1: Multiply: \(3\times 0^{2}+10\sin\left(0-2\right)\) - step2: Remove 0: \(3\times 0^{2}+10\sin\left(-2\right)\) - step3: Calculate: \(3\times 0+10\sin\left(-2\right)\) - step4: Multiply the numbers: \(0+10\sin\left(-2\right)\) - step5: Calculate: \(0+10\left(-\sin\left(2\right)\right)\) - step6: Rewrite the expression: \(0-10\sin\left(2\right)\) - step7: Remove 0: \(-10\sin\left(2\right)\) Calculate or simplify the expression \( 3*(3)^{2} + 10*\sin(2*(3) - 2) \). Calculate the value by following steps: - step0: Calculate: \(3\times 3^{2}+10\sin\left(2\times 3-2\right)\) - step1: Multiply the numbers: \(3\times 3^{2}+10\sin\left(6-2\right)\) - step2: Subtract the numbers: \(3\times 3^{2}+10\sin\left(4\right)\) - step3: Calculate: \(3^{3}+10\sin\left(4\right)\) - step4: Evaluate the power: \(27+10\sin\left(4\right)\) - step5: Calculate the trigonometric value: \(19.431975\) Calculate or simplify the expression \( 3*(-1)^{2} + 10*\sin(2*(-1) - 2) \). Calculate the value by following steps: - step0: Calculate: \(3\left(-1\right)^{2}+10\sin\left(2\left(-1\right)-2\right)\) - step1: Simplify: \(3\left(-1\right)^{2}+10\sin\left(-2-2\right)\) - step2: Subtract the numbers: \(3\left(-1\right)^{2}+10\sin\left(-4\right)\) - step3: Evaluate the power: \(3\times 1+10\sin\left(-4\right)\) - step4: Calculate: \(3\times 1+10\left(-\sin\left(4\right)\right)\) - step5: Multiply: \(3+10\left(-\sin\left(4\right)\right)\) - step6: Rewrite the expression: \(3-10\sin\left(4\right)\) - step7: Calculate the trigonometric value: \(10.568025\) Calculate or simplify the expression \( 3*(2)^{2} + 10*\sin(2*(2) - 2) \). Calculate the value by following steps: - step0: Calculate: \(3\times 2^{2}+10\sin\left(2\times 2-2\right)\) - step1: Multiply the numbers: \(3\times 2^{2}+10\sin\left(4-2\right)\) - step2: Subtract the numbers: \(3\times 2^{2}+10\sin\left(2\right)\) - step3: Multiply the terms: \(12+10\sin\left(2\right)\) - step4: Calculate the trigonometric value: \(21.092974\) Here are the calculated values of \( f'(c) \) at several points in the interval \((-2, 3)\): 1. \( c = -1 \): \[ f'(-1) \approx 10.568 \] 2. \( c = 0 \): \[ f'(0) \approx -9.093 \] 3. \( c = 1 \): \[ f'(1) \approx 3 \] 4. \( c = 2 \): \[ f'(2) \approx 21.093 \] 5. \( c = 3 \): \[ f'(3) \approx 19.432 \] Now, we can see that \( f'(-1) \) is greater than \( 8.614 \), and \( f'(0) \) is less than \( 8.614 \). This indicates that there is at least one root in the interval \((-1, 0)\). Next, we can narrow down the search for \( c \) in the interval \((-1, 0)\) by evaluating \( f' \) at a midpoint, say \( c = -0.5 \). Let's calculate \( f'(-0.5) \). Calculate the value by following steps: - step0: Calculate: \(3\left(-0.5\right)^{2}+10\sin\left(2\left(-0.5\right)-2\right)\) - step1: Multiply the numbers: \(3\left(-0.5\right)^{2}+10\sin\left(-1-2\right)\) - step2: Subtract the numbers: \(3\left(-0.5\right)^{2}+10\sin\left(-3\right)\) - step3: Convert the expressions: \(3\left(-\frac{1}{2}\right)^{2}+10\sin\left(-3\right)\) - step4: Calculate: \(3\left(-\frac{1}{2}\right)^{2}+10\left(-\sin\left(3\right)\right)\) - step5: Multiply the terms: \(\frac{3}{4}+10\left(-\sin\left(3\right)\right)\) - step6: Rewrite the expression: \(\frac{3}{4}-10\sin\left(3\right)\) - step7: Reduce fractions to a common denominator: \(\frac{3}{4}-\frac{10\sin\left(3\right)\times 4}{4}\) - step8: Transform the expression: \(\frac{3-10\sin\left(3\right)\times 4}{4}\) - step9: Multiply the terms: \(\frac{3-40\sin\left(3\right)}{4}\) - step10: Calculate the trigonometric value: \(-0.6612\)