To determine the intervals where the function $f(x) = \frac{1}{3}x^3 - 3x^2 + 9x + 26$ is increasing or decreasing, we need to find the first derivative $f'(x)$ and analyze its sign.

Step 1: Find the first derivative

Given: $f(x) = \frac{1}{3}x^3 - 3x^2 + 9x + 26$ the first derivative $f'(x)$ is: $f'(x) = x^2 - 6x + 9$

Step 2: Set $f'(x) = 0$ and solve for $x$

To find the critical points, we set $f'(x) = 0$: $x^2 - 6x + 9 = 0$ This factors as: $(x - 3)^2 = 0$ Thus, $x = 3$ is the only critical point.

Step 3: Test intervals around the critical point $x = 3$

We check the sign of $f'(x)$ in the intervals $(-\infty, 3)$ and $(3, \infty)$.

1. For $x < 3$, choose $x = 2$: $f'(2) = 2^2 - 6(2) + 9 = 4 - 12 + 9 = 1 > 0$ So, $f(x)$ is increasing on ( (-\infty, 3)