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: Find the derivative of \( f(x) \)
\[ f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3). \]
Step 2: Solve \( f'(x)<0 \)
Factorize: \[ f'(x)<0 \implies 4x^2(x - 3)<0. \] The critical points are \( x = 0 \) and \( x = 3 \). Analyze the sign of \( f'(x) \) in intervals: \[ (-\infty, 0), (0, 3), \text and (3, \infty). \]
Step 3: Determine intervals
\( f'(x)<0 \) on \( (-\infty, 0) \cup (0, 3) \).
Conclusion: The function is strictly decreasing on \( (-\infty, 0) \cup (0, 3) \).