Let's analyze each of the given problems step by step. If you'd like a detailed breakdown of any particular step, let me know.

---

26. $f(x) = x^3 - 6x^2 + 9x$

1. Derivative: $f'(x) = 3x^2 - 12x + 9$

2. Critical Points: Solve $f'(x) = 0$: $3x^2 - 12x + 9 = 0$ Divide by 3: $x^2 - 4x + 3 = 0$ Factorize: $(x - 3)(x - 1) = 0$ Critical points: $x = 1, x = 3$.

3. Second Derivative Test: $f''(x) = 6x - 12$

   • At $x = 1$: f''(1) = 6(1) - 12 = -6 \quad \text{(concave down, maximum at x = 1)}.
   • At $x = 3$: f''(3) = 6(3) - 12 = 6 \quad \text{(concave up, minimum at x = 3)}.

4. Max/Min Values:

   • Maximum at $x = 1$: $f(1) = 1^3 - 6(1)^2 + 9(1) = 4$.
   • Minimum at $x = 3$: $f(3) = 3^3 - 6(3)^2 + 9(3) = 0$.

---

27. $f'(x) = 4x^3 - 12x + 8$

1. Critical Points: Solve $f''(x) = 0$ to find possible inflection points or changes in concavity. $f''(x) = 12x^2 - 12 = 0$ Factorize: $12(x^2 - 1) = 0 \implies x = \pm 1$

   Analyze $f''(x)$ around $x = \pm 1$:

   • $x < -1$: $f''(x) > 0$ (concave up).
   • $-1 < x < 1$: $f''(x) < 0$ (concave down).
   • $x > 1$: $f''(x) > 0$ (concave up).

   Points of inflection: $x = -1, 1$.

---

28. Fungsi $f(x)$ memiliki nilai maksimum lokal

To determine whether a function has a local maximum, we examine its derivative and apply the First Derivative Test or Second Derivative Test.

---

29. $f(x) = x^4 - 2x^2 + x$

1. Derivative: $f'(x) = 4x^3 - 4x + 1$

2. Critical Points: Solve $f'(x) = 0$: $4x^3 - 4x + 1 = 0$ This is a cubic equation. Numerical or approximation methods (like Newton's method) can solve this.

3. Second Derivative: $f''(x) = 12x^2 - 4$

   Solve $f''(x) = 0$: $12x^2 - 4 = 0 \implies x^2 = \frac{1}{3} \implies x = \pm \frac{\sqrt{3}}{3}$

   These are points of inflection.

---

30. $f(x) = -x^3 + 3x^2 - 5x + 7$

1. Derivative: $f'(x) = -3x^2 + 6x - 5$

2. Critical Points: Solve $f'(x) = 0$: $-3x^2 + 6x - 5 = 0$ Multiply by $-1$: $3x^2 - 6x + 5 = 0$ Discriminant: $\Delta = (-6)^2 - 4(3)(5) = 36 - 60 = -24$ No real solutions (no critical points in the real domain).

3. Concavity: $f''(x) = -6x + 6$ Solve $f''(x) = 0$: $-6x + 6 = 0 \implies x = 1$

   • $x < 1$: $f''(x) > 0$ (concave up).
   • $x > 1$: $f''(x) < 0$ (concave down).

---

Questions:

1. Would you like a detailed solution to any specific function?
2. Should we graph any of these functions to visualize their behavior?
3. Do you want help solving higher-order equations numerically?
4. Would you like to explore concavity and inflection points further?
5. Should we discuss local and global extrema in more depth?

Tip: Always analyze the second derivative to understand the shape of the function and verify extrema effectively.