Find the inlorvals on which \( (x) \) is increasing and the intervals on which \( I(x) \) is decreasing. Then sketch the graph. Add hcrizontal tangent lines. \[ f(x)=2 x^{4}-100 x^{2} \] Compute the derivative of \( \mathrm{I}(\mathrm{x}) \). \[ f^{\prime}(x)=8 x^{3}-200 x \] Selad the cortect choice below and if nocossary fill in the answor box to complete yout choice. A. The function is incroasing on \( (-5,0),(5, \infty) \). (Type your answer using interval nolation. Use \& comma to separate answers as needed.) B. The fundion is never increasing. Select the correct choice bolow and, if necessary, fill in the enswer box to complele your choico. A. Tho function is decreasing on \( \square \) . (Type you answer using intarval notation. Uso a comma to separale answers as needed) B. There function is nover decieasing.

To find the intervals where \( f(x) = 2x^4 - 100x^2 \) is increasing or decreasing, we first compute the derivative: \[ f'(x) = 8x^3 - 200x = 8x(x^2 - 25) = 8x(x - 5)(x + 5). \] Setting this equal to zero gives us \( x = -5, 0, 5 \). Next, we analyze the sign of \( f'(x) \): - On the interval \( (-\infty, -5) \): Choose \( x = -6 \), \( f'(-6) < 0 \) (decreasing). - On the interval \( (-5, 0) \): Choose \( x = -1 \), \( f'(-1) > 0 \) (increasing). - On the interval \( (0, 5) \): Choose \( x = 1 \), \( f'(1) > 0 \) (increasing). - On the interval \( (5, \infty) \): Choose \( x = 6 \), \( f'(6) < 0 \) (decreasing). In summary, A. The function is increasing on \( (-5, 0) \) and \( (0, 5) \). B. The function is decreasing on \( (-\infty, -5) \) and \( (5, \infty) \). To sketch the graph, plot \( f(x) \) and indicate \( x = -5, 0, 5 \) with horizontal tangent lines at those points. --- The function's behavior across its critical points can be contextually fascinating! Did you know that polynomials like this one show a balance between degrees and coefficients that give rise to such beautiful curves? Their symmetries reveal insights into real-world phenomena, like the forces acting on structures or optimizing resources. Next time you see a graph, think of the story it tells! If you want to dive deeper into polynomial functions and their implications across various fields—from engineering to economics—there's an entire world of resources waiting for you! Look for books on calculus or mathematical modeling; they can be quite enlightening and fun!