Find shortest distance from origin to the cubic y=x^3 −10x^2 +27x−18.


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Question Number 54983 by ajfour last updated on 15/Feb/19

$F i n d s h o r t e s t d i s t a n c e f r o m o r i g i n$ $t o t h e c u b i c y = x^{3} - 10 x^{2} + 27 x - 18 .$
Commented by mr W last updated on 15/Feb/19

$y o u c a n a p p l y y o u r s o l u t i o n m e t h o d f o r$ $q u i n t i c e q u a t i o n s t o s o l v e t h i s q u e s t i o n .$
Commented by ajfour last updated on 15/Feb/19

$i t w a s n o t g e n e r a l s i r, i h a d$ $c o m p a r e d c o e f f i c i e n t s o f t w o$ $q u a d r a t i c e q u a t i o n s i n f .$ $y = (x^{2} + α x + f) (x^{3} + {R x}^{2} + p x + q) .$
	Answered by mr W last updated on 16/Feb/19

	Commented by mr W last updated on 16/Feb/19

	$c u r v e y = f (x)$ $P (x, y)$ $O P i s n o r m a l o f t h e c u r v e :$ $\frac{y}{x} = - \frac{1}{y^{'}}$ $\Rightarrow {y y}^{'} + x = 0$ $w i t h y = x^{3} - 10 x^{2} + 27 x - 18$ $y^{'} = 3 x^{2} - 20 x + 27$ $\Rightarrow (x^{3} - 10 x^{2} + 27 x - 18) (3 x^{2} - 20 x + 27) + x = 0$ $\Rightarrow 3 x^{5} - 50 x^{4} + 308 x^{3} - 864 x^{2} + 1090 x - 486 = 0$ $\Rightarrow x_{m i n} \approx 0.990296$ $\Rightarrow y \approx - 0.0977$ $\Rightarrow {O P}_{m i n} = d_{m i n} \approx 0.9905$
	Commented by mr W last updated on 16/Feb/19

	Commented by mr W last updated on 16/Feb/19

	$i s t h e r e a w a y w i t h o u t s o l v i n g t h e$ $q u i n t i c e q u a t i o n ?$
	Commented by mr W last updated on 16/Feb/19

	$D = d^{2} = x^{2} + y^{2}$ $\frac{d D}{d x} = 2 x + 2 {y y}^{'} = 0$ $\Rightarrow x + {y y}^{'} = 0 \Rightarrow t h i s l e a d s t o q u i n t i c e q n .$
	Commented by MJS last updated on 16/Feb/19

	$no Sir .$ $we can also go this way :$ $\frac{d}{d x} [f {(x)}^{2} + x^{2}] = 0$ $which also leads to a quintic with at least$ $no easy exact solution$
	Commented by ajfour last updated on 16/Feb/19

	$B e a u t i f u l d e m o n s t r a t i o n S i r,$ $t h a n k s a l o t . i t h i n k w h a t s o e v e r$ $w e d o {i t}^{'} l l l a n d o n s o l v i n g a d e g r e e 5 .$
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