1.96 / 5.88 points\ndetermine the open intervals on which the function is increasing, decreasing, or…

1.96 / 5.88 points\ndetermine the open intervals on which the function is increasing, decreasing, or constant. (enter your answers using interval notation. if an answer does not exist, enter dne.)\n$f(x)=-\\frac{1}{2} x^{3}$

1.96 / 5.88 points\ndetermine the open intervals on which the function is increasing, decreasing, or constant. (enter your answers using interval notation. if an answer does not exist, enter dne.)\n$f(x)=-\\frac{1}{2} x^{3}$

Answer

Explanation:

Step1: Find the derivative of the function

The function is ( f(x)=-\frac{1}{2}x^{3} ). Using the power rule ( (x^{n})^\prime = nx^{n - 1} ), the derivative ( f^\prime(x)=-\frac{1}{2}\times3x^{2}=-\frac{3}{2}x^{2} ).

Step2: Analyze the sign of the derivative

Since ( x^{2}\geq0 ) for all real ( x ), then ( -\frac{3}{2}x^{2}\leq0 ) for all real ( x ). The equality ( -\frac{3}{2}x^{2} = 0 ) holds when ( x = 0 ). For ( x\neq0 ), ( f^\prime(x)<0 ).

Answer:

increasing: DNE; decreasing: ( (-\infty,\infty) ); constant: DNE