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.
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.
Show Hint
To determine increasing and decreasing intervals, solve \( f'(x) = 0 \) and test the sign of \( f'(x) \) in each interval.
Updated On: Mar 3, 2025
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Solution and Explanation
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 \( (-2,1) \cup (3,\infty) \).
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