Penyelesai persamaan padu khusus dengan punca sebenar dan kompleks, langkah kaedah Cardano, grafik padu dan contoh yang berfungsi.
Kalkulator Mata Pusing
Masukkan pekali polinomial anda di atas dan klik "Cari Titik Pusing" untuk melihat keputusan.Turning points (also called local extrema) are locations where a cubic function changes direction — from increasing to decreasing (local maximum) or from decreasing to increasing (local minimum). They are found by solving the first derivative equation: f'(x) = 3ax² + 2bx + c = 0, which is a quadratic in x.
The discriminant of the first derivative, D = 4b² − 12ac, determines whether turning points exist. When D > 0, the cubic has two turning points (one max, one min). When D = 0, there is a single horizontal inflection (a saddle point). When D < 0, the cubic is monotonic with no turning points — it always increases or always decreases.
Turning points are critical for optimization, graphing, and understanding function behavior. The vertical distance between turning points determines the “amplitude” of the cubic's wiggle, and their x-coordinates define the boundaries between increasing and decreasing intervals. Engineers use them to find maximum stress, peak voltage, or optimal production levels.
Finding the local maximum of a cubic revenue model reveals the optimal production quantity for maximum profit.
Peak stress and deflection in structural components often occur at turning points of the governing cubic equation.
Population models with cubic dynamics use turning points to identify carrying capacities and extinction thresholds.
Turning points are where f'(x)=0 (direction changes). Inflection points are where f''(x)=0 (concavity changes). They are different.
When 4b² − 12ac is negative, the cubic is monotonic. Don't try to force turning points that don't exist.
Finding the x-values isn't enough. Use the second derivative test: f''(x) > 0 means minimum, f''(x) < 0 means maximum.
| Derivative | f'(x) = 3ax² + 2bx + c = 0 |
| D > 0 | Two turning points (1 max + 1 min) |
| D = 0 | Saddle point (horizontal inflection) |
| D < 0 | No turning points (monotonic) |
| Classification | Use f''(x) to identify max vs. min |
Tidak, kubik biasanya mempunyai sama ada betul-betul dua titik pusingan, atau tiada langsung (ia meningkat atau berkurangan).
Jika titik pusingan terletak tepat pada paksi-x, persamaan itu mempunyai punca "berulang" atau "berganda" pada koordinat itu!
Tidak, tetapi ia sangat membantu dalam menggambarkan geometri.
Diskriminasi terbitan pertama (kuadrat) menentukan ini. Jika 4b² - 12ac > 0, kubik mempunyai dua titik pusingan; jika tidak ia tidak mempunyai.
ya. Jika kedua-dua titik pusingan berada di atas paksi-x (atau kedua-duanya di bawah), kubik hanya mempunyai satu punca sebenar. Ini betul-betul kes di mana akar kompleks muncul.
