«11C» 40. Hausaufgabe «PDF», «POD»

0.0.1 ↑ 40. Hausaufgabe

0.0.1.1 ↑ Selbstgestellte Aufgabe

f(x) = ax^3 + bx^2 + cx + d$f\left(x\right)=a{x}^3+b{x}^2+cx+d$ berührt die Gerade g(x) = 12x + 13$g\left(x\right)=12x+13$ in A(-1, 1)$A\left(-1,1\right)$ und hat in B(1, 5)$B\left(1,5\right)$ eine waagrechte Tangente.

{} f(-1) = 1 = -a + b - c + d; \\ {} \qquad \Rightarrow d = 1 + a - b + c; \\ {} f(1) = 5 = a + b + c + d = a + b + c + 1 + a - b + c; \\ {} \qquad \Rightarrow 4 = 2a + 2c; \\ {} \qquad \Rightarrow c = 2 - a; \\ {} f'(-1) = 0 = 3a + 2b + c = 3a + 2b + 2 - a; \\ {} \qquad \Rightarrow -2 = 2a + 2b; \\ {} \qquad \Rightarrow b = -1 - a; \\ {} f'(-1) = 12 = 3a - 2b + c = 3a + 2 + 2a + 2 - a = 4a + 4; \\ {} \qquad \Rightarrow 8 = 4a; \\ {} \qquad \Rightarrow a = 2; \\ {} \qquad \Rightarrow b = -1 - 2 = -3; \\ {} \qquad \Rightarrow c = 2 - 2 = 0; \\ {} \qquad \Rightarrow d = 1 + 2 + 3 = 6; \\ {} \qquad \Rightarrow f(x) = 2x^3 - 3x^2 + 6;$f\left(-1\right)=1=-a+b-c+d;\Rightarrow d=1+a-b+c;f\left(1\right)=5=a+b+c+d=a+b+c+1+a-b+c;\Rightarrow 4=2a+2c;\Rightarrow c=2-a;f^{\prime }\left(-1\right)=0=3a+2b+c=3a+2b+2-a;\Rightarrow -2=2a+2b;\Rightarrow b=-1-a;f^{\prime }\left(-1\right)=12=3a-2b+c=3a+2+2a+2-a=4a+4;\Rightarrow 8=4a;\Rightarrow a=2;\Rightarrow b=-1-2=-3;\Rightarrow c=2-2=0;\Rightarrow d=1+2+3=6;\Rightarrow f\left(x\right)=2x^3-3x^2+6;$