Find the range of all values of P.

Oct 2019
9
4
India
Find the range of all possible values of B if the graph of
P(x) = 12x⁴ - 5x³ -38x² + Bx + 6
crosses the x-axis between 0 and -1

[I have concluded some points from the problem. I still need help finding the range . My working is in the attachments]
 

Attachments

Jun 2013
1,096
573
Lebanon
without using calculus?

is \(\displaystyle B\) an integer?
 
Jun 2013
1,096
573
Lebanon
If \(\displaystyle p(x)=12x^{4 }-5x^3-38x^2+b x+6\) has a root \(\displaystyle r\), \(\displaystyle -1\leq r <0\) then \(\displaystyle b \geq -15\)

To see this , write \(\displaystyle p(r)=0\) and show that

\(\displaystyle b+15=-\frac{(1+r) }{r}g(r)\)

where \(\displaystyle g(x)=6-21 x-17 x^2+12 x^3\)

1) \(\displaystyle g(x)\) has a unique root \(\displaystyle \alpha\) where \(\displaystyle -1<\alpha < -\frac{2}{3}\)

2) If \(\displaystyle r>\alpha\) we are done. Otherwise

\(\displaystyle -2<g(r)<0\) and \(\displaystyle 0<-\frac{(1+r) }{r}<\frac{1}{2}\)

From this it follows that \(\displaystyle b>-16\) and again we get \(\displaystyle b \geq -15\) since \(\displaystyle b\) is an integer

Conversely, show that if \(\displaystyle b \geq -15\) then

\(\displaystyle p(x)=12x^{4 }-5x^3-38x^2+b x+6\) has a root \(\displaystyle r\), \(\displaystyle -1\leq r <0\)
 
  • Like
Reactions: topsquark