Questions: Find the intervals where f is increasing and decreasing: f(x) = -1/5 x^5 - 17/4 x^4 - 70/3 x^3 + 6

△ Determine intervals where \( f(x) \) is increasing and decreasing. ○ Find the derivative \( f'(x) \) ☼ The derivative is \( f'(x) = -x^4 - 17x^3 - 70x^2 \). Calculating the derivative of each term: \(-\frac{1}{5}(5x^4) = -x^4\), \(-\frac{17}{4}(4x^3) = -17x^3\), \(-\frac{70}{3}(3x^2) = -70x^2\), and the derivative of the constant 6 is 0. ○ Find critical points ☼ Critical points are \( x = -10, -7, 0 \). Setting \( f'(x) = 0 \) gives \(-x^2(x^2 + 17x + 70) = 0\). Solving \( x^2 + 17x + 70 = 0 \) by factorization yields \((x+7)(x+10) = 0\), leading to \( x = -7 \) and \( x = -10 \). ○ Analyze intervals ☼ The function is increasing on \((-10, -7)\) and decreasing on \((-\infty, -10) \cup (-7, \infty)\). Testing values in each interval: \( f'(-11) < 0 \), \( f'(-8) > 0 \), \( f'(-1) < 0 \), \( f'(1) < 0 \). The sign of \( f'(x) \) determines the behavior of \( f(x) \). ✧ The function is increasing on \((-10, -7)\) and decreasing on \((-\infty, -10) \cup (-7, \infty)\). ☺ The function is increasing on the interval $(-10, -7)$. The function is decreasing on the intervals $(-\infty, -10)$ and $(-7, \infty)$.