. جد نقاط النهايات العظمى والصغوى ان وجدت \( \quad f(x)=x^{3}(-4+x) \) لكمن

Step 1 ：1. \(f'(x) = \frac{d}{dx}(x^3(-4+x))\)

Step 2 ：2. \(f'(x) = 3x^2(-4+x) - x^3(1)\)

Step 3 ：3. \(f'(x) = x^2(3x-4) - x^3\)

Step 4 ：4. \(f'(x) = x^2(3x^2-4x-x^2)\)

Step 5 ：5. \(f'(x) = x^2(3x^2-4x-x^2)\)

Step 6 ：6. Set \(f'(x) = 0 \) and solve for \(x\).

Step 7 ：7. \(x = 0\) or \(3x^2-4x-x^2=0\)

Step 8 ：8. \(x=0\) or \(2x^2-4x=0\)

Step 9 ：9. \(x=0\) or \(x(2x-4)=0\)

Step 10 ：10. \(x=0\), \(x=2\)

Step 11 ：11. Determine if \(f(x)\) has a local maximum or minimum at each critical point.

Step 12 ：12. \(f''(x) = \frac{d^2}{dx^2}f(x)\)

Step 13 ：13. \(f''(x) = 2x(3x-4) + x^2(3)\)

Step 14 ：14. Test each critical point in \(f''(x)\) for concavity.

Step 15 ：15. At \(x=0\): \(f''(0) = 0(0-4) + 0^2(3) = 0\).

Step 16 ：16. At \(x=2\): \(f''(2) = 2(6-4) + 4(3) = 12\).

Step 17 ：17. Identify the local extrema: There is a local minimum at \(x=2\), but the function has no local maximum.