Question

Find the interval in which the function \( f(x) = x^4 - 4x^3 + 10 \) is strictly decreasing.

Hint:

Use the derivative sign test to find intervals of increase or decrease.

Answer 1

Step 1: Calculate the derivative of \( f(x) \).
\[f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3).\]
Step 2: Solve the inequality \( f'(x)<0 \).
Factor the expression:
\[f'(x)<0 \implies 4x^2(x - 3)<0.\]Identify critical points at \( x = 0 \) and \( x = 3 \). Examine the sign of \( f'(x) \) across the intervals:
\[(-\infty, 0), (0, 3), \text and (3, \infty).\]
Step 3: Define the intervals where \( f'(x)<0 \).
\( f'(x)<0 \) on \( (-\infty, 0) \cup (0, 3) \).
Conclusion: The function \( f(x) \) is strictly decreasing on \( (-\infty, 0) \cup (0, 3) \).