How-do-you-make-a-curve-y-ax-3-bx-2-cx-d-with-a-critical-point-of-1-0-and-2-27-

Question Number 187053 by horsebrand11 last updated on 13/Feb/23

H o w d o y o u m a k e a c u r v e

y = {a x}^{3} + {b x}^{2} + c x + d w i t h a c r i t i c a l

p o i n t o f (1, 0) a n d (- 2, 27) ?

Answered by cortano12 last updated on 13/Feb/23

\Rightarrow 3 {a x}^{2} + 2 b x + c = k (x - 1) (x + 2)

\Rightarrow 3 {a x}^{2} + 2 b x + c = {k x}^{2} + k x - 2 k

\Rightarrow {\begin{cases} k = 3 a = 2 b; b = \frac{3}{2} a \\ c = - 2 k = - 6 a \end{cases}

∵ y = {a x}^{3} + \frac{3}{2} {a x}^{2} - 6 a x + d

\Rightarrow - \frac{7}{2} a + d = 0 \Rightarrow d = \frac{7}{2} a

\Rightarrow 10 a + d = 27 \Rightarrow \frac{27}{2} a = 27; a = 2

{\begin{cases} d = 7 \\ c = - 12 \\ b = 3 \end{cases} \Rightarrow y = 2 x^{3} + 3 x^{2} - 12 x + 7