Question

1. use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant.\\( f(x) = 2x^4 - 4x^2 \\)\

a) decreasing: \\( (-infty, -1) cup (1, infty) \\); increasing: \\( (-1, 1) \\)\
b) decreasing: \\( (-infty, 0) \\); increasing \\( (0, infty) \\)\
c) decreasing: \\( (-infty, -1) cup (0, 1) \\); increasing: \\( (-1, 0) cup (1, infty) \\)\
d) decreasing: \\( (0, 0) \\); increasing: \\( (0, infty) \\)\
e) decreasing: \\( (-1, 0) cup (1, infty) \\); increasing: \\( (-infty, -1) cup (0, 1) \\)

Explanation:

Step1: Find critical points

Take derivative: $f'(x) = 8x^3 - 8x = 8x(x^2-1) = 8x(x-1)(x+1)$. Set to 0: $x=-1, 0, 1$.

Step2: Test interval sign

1. $(-\infty,-1)$: $f'(x) <0$, function decreasing
2. $(-1,0)$: $f'(x) >0$, function increasing
3. $(0,1)$: $f'(x) <0$, function decreasing
4. $(1,\infty)$: $f'(x) >0$, function increasing

Answer:

c) Decreasing: $(-\infty, -1) \cup (0, 1)$; Increasing: $(-1, 0)\cup (1, \infty)$