The derivative of the function f is defined by f^(')(x)=x^(3)-5+2sin(2x-5). Find the x values, if any, in the interval -0.5

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The derivative of the function  f is defined by  f^(')(x)=x^(3)-5+2sin(2x-5). Find the  x values, if any, in the interval  -0.5 < x < 2.5 where the function  f has a relative maximum. You may use a calculator and round all values to 3 decimal places. Answer:  x=

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Find Critical Points: To find the relative maximum of the function f, we need to find the critical points of f^(')(x) in the given interval. Critical points occur where the derivative is zero or undefined. The derivative f^(')(x) = x^(3) - 5 + 2sin(2x - 5) is defined for all x, so we only need to find where it is zero.Solve for x: Set the derivative equal to zero and solve for x: x^(3) - 5 + 2sin(2x - 5) = 0 This equation is not easily solvable algebraically, so we will use a calculator to find the approximate values of x in the interval -0.5 < x < 2.5.Graph Function: Using a calculator, we graph the function f^(')(x) = x^(3) - 5 + 2sin(2x - 5) and look for points where the graph crosses the x-axis within the interval -0.5 < x < 2.5.Check Sign Change: After graphing, we find that the derivative crosses the x-axis at approximately x = 1.047. We need to check if this is a relative maximum by looking at the sign of the derivative before and after this point.Verify Relative Maximum: We test the sign of the derivative around x = 1.047. If the derivative changes from positive to negative at x = 1.047, then it is a relative maximum.Using the calculator, we find that the derivative is positive just before x = 1.047 and negative just after x = 1.047. Therefore, x = 1.047 is a relative maximum.

The derivative of the function $f$ is defined by $f^{\prime}(x)=x^{3}-5+2 \sin (2 x-5)$ . Find the $x$ values, if any, in the interval \( -0.5

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The derivative of the function f f f is defined by f′(x)=x3−5+2sin⁡(2x−5) f^{\prime}(x)=x^{3}-5+2 \sin (2 x-5) f′(x)=x3−5+2sin(2x−5). Find the x x x values, if any, in the interval \( -0.5

The derivative of the function $f$ is defined by $f^{\prime}(x)=x^{3}-5+2 \sin (2 x-5)$ . Find the $x$ values, if any, in the interval \( -0.5