This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Haya kelvinobenj, twende na swali hili la curve. Equation ya curve ni y = 3x^2 - x^3. a) Kokotoa coordinates za turning points za curve. Step 1: Tafuta derivative ya kwanza ya y kwa heshima ya x. y = 3x^2 - x^3 (dy)/(dx) = 6x - 3x^2 Step 2: Weka derivative ya kwanza sawa na sifuri ili kupata maadili ya x kwenye turning points. 6x - 3x^2 = 0 3x(2 - x) = 0 Hii inatoa solutions mbili: 3x = 0 x = 0 2 - x = 0 x = 2 Step 3: Badilisha maadili ya x kwenye equation asili ya curve ili kupata maadili yanayolingana ya y. Kwa x = 0: y = 3(0)^2 - (0)^3 = 0 Turning point ya kwanza ni (0, 0). Kwa x = 2: y = 3(2)^2 - (2)^3 = 3(4) - 8 = 12 - 8 = 4 Turning point ya pili ni (2, 4). Coordinates za turning points ni (0, 0) na (2, 4). b) Kokotoa aina ya stationary points. Step 1: Tafuta derivative ya pili ya y kwa heshima ya x. (d^2y)/(dx^2) = (d)/(dx)(6x - 3x^2) = 6 - 6x Step 2: Tathmini derivative ya pili kwa kila turning point. Kwa x = 0: (d^2y)/(dx^2) = 6 - 6(0) = 6 Kwa kuwa (d^2y)/(dx^2) > 0, point (0, 0) ni local minimum. Kwa x = 2: (d^2y)/(dx^2) = 6 - 6(2) = 6 - 12 = -6 Kwa kuwa (d^2y)/(dx^2) < 0, point (2, 4) ni local maximum. Aina ya stationary points ni: (0, 0) ni local minimum. (2, 4) ni local maximum. c) Kokotoa intercepts. Step 1: Tafuta y-intercept (ambapo x = 0). Badilisha x = 0 kwenye equation asili: y = 3(0)^2 - (0)^3 = 0 Y-intercept ni (0, 0). Step 2: Tafuta x-intercepts (ambapo y = 0). Weka y = 0 kwenye equation asili: 0 = 3x^2 - x^3 Toa x^2 kama common factor: 0 = x^2(3 - x) Hii inatoa solutions mbili: x^2 = 0 x = 0 3 - x = 0 x = 3 X-intercepts ni (0, 0) na (3, 0). d) Chora curve. Ili kuchora curve, tunatumia taarifa zilizopatikana: • Turning points: (0, 0) (minimum) na (2, 4) (maximum). • Intercepts: (0, 0) (y-intercept na x-intercept) na (3, 0) (x-intercept). • Tabia ya curve: Kama x -, y . Kama x , y -. Mchoro wa curve utaonyesha: 1. Curve inatoka juu kushoto (kutoka ). 2. Inashuka hadi kufikia local minimum kwenye (0, 0). 3. Kisha inapanda hadi kufikia local maximum kwenye (2, 4). 4. Baada ya hapo, inashuka tena, ikipita kwenye x-axis kwenye (3, 0). 5. Inaendelea kushuka kuelekea chini kulia (kuelekea -). Last free one today — make it count tomorrow, or type /upgrade for unlimited.