The derivative of the function f is defined by f^(')(x)=x^(2)+2x-3sin(3x). Find the x values, if any, in the interval -2.5

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The derivative of the function  f is defined by  f^(')(x)=x^(2)+2x-3sin(3x). Find the  x values, if any, in the interval  -2.5 < x < 3 where the function  f has a relative minimum. You may use a calculator and round all values to 3 decimal places. Answer:  x=

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Identify Critical Points: Identify the critical points of the function f by setting its derivative f'(x) equal to zero and solving for x. f'(x) = x^2 + 2x - 3sin(3x) = 0 This is a transcendental equation and may not have an algebraic solution. We will use a calculator to find the roots in the interval -2.5 < x < 3.Find Zero of Derivative: Use a graphing calculator or numerical methods to find the approximate values of x where f'(x) = 0 in the given interval. By graphing or using a numerical solver, we can find the x values that make the derivative zero. These x values are the potential points where f could have a relative minimum or maximum.Determine Relative Minimum: Determine whether each critical point is a relative minimum by using the second derivative test or the first derivative test. For the second derivative test, we would find f''(x) and evaluate it at the critical points. If f''(x) > 0, then f has a relative minimum at that point. If f''(x) < 0, then f has a relative maximum at that point. Alternatively, for the first derivative test, we would look at the sign of f'(x) before and after each critical point. If f'(x) changes from negative to positive at a critical point, then f has a relative minimum at that point.Calculate Second Derivative: Use the calculator to find the second derivative of f, if necessary, and evaluate it at the critical points found in Step 2. Since the second derivative of f involves the derivative of -3sin(3x), which is -9cos(3x), the second derivative f''(x) will be f''(x) = 2x + 2 - 9cos(3x). We can then use the calculator to evaluate f''(x) at each critical point to determine if it is a relative minimum.Report Minimum Values: Report the x values where f has a relative minimum, rounding to three decimal places as instructed. After evaluating the second derivative at the critical points or using the first derivative test, we can identify the x values where f has a relative minimum. These values should be within the interval -2.5 < x < 3.

The derivative of the function $f$ is defined by $f^{\prime}(x)=x^{2}+2 x-3 \sin (3 x)$ . Find the $x$ values, if any, in the interval \( -2.5

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The derivative of the function f f f is defined by f′(x)=x2+2x−3sin⁡(3x) f^{\prime}(x)=x^{2}+2 x-3 \sin (3 x) f′(x)=x2+2x−3sin(3x). Find the x x x values, if any, in the interval \( -2.5

The derivative of the function $f$ is defined by $f^{\prime}(x)=x^{2}+2 x-3 \sin (3 x)$ . Find the $x$ values, if any, in the interval \( -2.5