The derivative of the function \( f \) is defined by \( f^{\prime}(x)=x^{2}+4 \cos (2 x+2) \) for \( -2

Ask by Lewis Hardy. in the United States

Jan 27,2025

Simplify the expression by following steps: - step0: Solution: \(x^{2}+4\cos\left(2x+2\right)\) It seems that I cannot directly solve the equation numerically at this moment. However, I can guide you through the process of finding the roots using numerical methods or a graphing calculator. ### Step 6: Numerical Method You can use a numerical method such as the Newton-Raphson method or a graphing calculator to find the roots of the equation \( x^{2} + 4 \cos(2x + 2) = 0 \) in the interval \( -2 < x < 4 \). ### Step 7: Analyze the intervals 1. **Check endpoints**: Evaluate \( f^{\prime}(-2) \) and \( f^{\prime}(4) \) to see if they are positive or negative. 2. **Test intervals**: Choose test points in the intervals determined by the roots found in the previous step to see if \( f^{\prime}(x) \) is positive or negative. ### Step 8: Conclusion - If \( f^{\prime}(x) > 0 \) in an interval, then \( f \) is increasing in that interval. - If \( f^{\prime}(x) < 0 \), then \( f \) is decreasing. ### Example of Testing Intervals 1. If you find roots at \( x = a \) and \( x = b \), test points in the intervals \( (-2, a) \), \( (a, b) \), and \( (b, 4) \). 2. For each test point, substitute into \( f^{\prime}(x) \) to determine the sign. ### Final Step Once you have the roots and the signs of \( f^{\prime}(x) \) in each interval, you can conclude where \( f \) is increasing. If you have access to a graphing calculator or software, you can plot \( f^{\prime}(x) \) to visually identify the intervals where it is positive. Would you like me to assist you with any specific calculations or further steps?