3.3 Lösning 8a
För att kunna derivera utvecklas \( \, f(x) \, \) till ett polynom:
\[ f(x) \, = \, x^2 \, (x + 1) \, (2\,x + 5) + 1 \, = \, (x^3 + x^2) \, (2\,x + 5) + 1 \, = \]
- \[ \quad = \, 2\,x^4 + 5\,x^3 + 2\,x^3 + 5\,x^2 + 1 \, = \, 2\,x^4 + 7\,x^3 + 5\,x^2 + 1 \]
Vi deriverar \( \, f(x) \, \) två gånger:
\[\begin{array}{rcl} f(x) & = & 2\,x^4 + 7\,x^3 + 5\,x^2 + 1 \\ f'(x) & = & 8\,x^3 + 21\,x^2 + 10\,x \\ f''(x) & = & 24\,x^2 + 42\,x + 10 \end{array}\]
Derivatans nollställen:
\[\begin{array}{rcl} 8\,x^3 + 21\,x^2 + 10\,x & = & 0 \\ x\,(8\,x^2 + 21\,x + 10) & = & 0 \\ x_1 & = & 0 \\ 8\,x^2 + 21\,x + 10 & = & 0 \\ x^2 + \frac{21}{8}\,x + \frac{10}{8} & = & 0 \\ x^2 + 2,625\,x + 1,25 & = & 0 \\ x_{2,3} & = & -1,3125 \pm \sqrt{1,7227 - 1,25} \\ x_{2,3} & = & -1,3125 \pm 0,6875 \\ x_2 & = & - 0,625 \\ x_3 & = & - 2 \end{array}\]
