Question

Find the interval in which the function \[ f(x) = \frac{3}{10}x^4 - \frac{4}{5}x^3 - 3x^2 + \frac{36}{5}x + 11 \] is (A) increasing, (B) decreasing.

Hint:

To determine increasing and decreasing intervals, solve \( f'(x) = 0 \) and test the sign of \( f'(x) \) in each interval.

Answer 1

Compute the derivative: \[ f'(x) = \frac{6}{5}x^3 - \frac{12}{5}x^2 - 6x + \frac{36}{5}. \] Simplify: \[ f'(x) = \frac{6}{5}(x^3 - 2x^2 - 5x + 6). \] Solve \( f'(x) = 0 \) to find critical points: \[ x^3 - 2x^2 - 5x + 6 = 0. \]  Test intervals \( (-\infty, -2) \), \( (-2,1) \), \( (1,3) \), \( (3,\infty) \) by substituting values in \( f'(x) \):

• \( f'(x) > 0 \): Function is increasing.
• \( f'(x) < 0 \): Function is decreasing.

Thus, \( f(x) \) is increasing on \( (-\infty, -2) \cup (1,3) \) and decreasing on \( (-2,1) \cup (3,\infty) \).