\hspace{-16}$For $\bf{(85)::}$ Given $\bf{ax^3+bx^2+cx+d = a(x-1)^2\cdot (x-\gamma).........(1)}$\\\\\\ Where $\bf{a>0}$ and $\bf{\gamma<0}$, Now put $\bf{x=1}$ on both side, we get\\\\\\ $\bf{\Rightarrow a+b+c+d = 0\Rightarrow c=-\left(a+b+d\right)}$\\\\\\ Now Diif. equation $\bf{(1)}$ both side w. r to $\bf{x\;,}$ we get\\\\\\ $\bf{\Rightarrow 3ax^2+2bx+c = a\left\{(x-1)^2+2(x-1)(x-\gamma)\right\}}$\\\\\\ Now put $\bf{x=1}$ on both side, we get\\\\\\ $\bf{\Rightarrow 3a+2b+c = 0\Rightarrow 3a+2b-a-b-d = 0\Rightarrow \boxed{\boxed{\bf{2a+b=d}}}}$\\\\\\ (Using $\bf{c=-\left(a+b+d\right)}$)\\\\\\ Now Using Product of Roots for equation $\bf{(1)\;,}$ We get\\\\\\ $\bf{\Rightarrow \gamma = -\frac{d}{a}\Rightarrow d=-\gamma \cdot a >0}$\\\\\\ So $\bf{d=2a+b>0}$\\\\\\ So $\bf{\boxed{\boxed{\bf{d+a= 2a+b+b>0\Rightarrow 3a+b>0}}}}$\\\\\\ bcz $\bf{a\;,d >0}$
6 Answers
Manish Shankar
·2014-08-13 00:28:21
The graph will look like this
Which shows it has 5 critical point
shivendu gupta thank you vey much.Upvote·0· Reply ·2014-08-13 09:23:23
Amartya Kumar Mistry
·2014-08-11 23:44:07
drawing the graph i am getting option c for the q. 83
shivendu gupta given answer is d.
Amartya Kumar Mistry
·2014-08-11 23:55:59
for q. 89 ans is option 2
shivendu gupta given answer is d here too.
Himanshu Giria
·2014-08-13 04:38:56
84 .(2)
Himanshu Giria Where are the other options for85??
Himanshu Giria 86 is marked correct as4
Himanshu Giria Sorry88not86
shivendu gupta this is correct,wil you please tell me how to solve?
shivendu gupta for ques. 85 options are :
(c) 3a-b>o,d=3a+2b
(d) none
Himanshu Giria
·2014-08-13 04:50:46
In 90 i guess α can have any real value so I guess all the4options are correct
...
shivendu gupta given ans. is 3
Sahil Jain alpha cannot be negative since if its is negative the funtion becomes logx/x which is not defined at x = o. So contradictory results appear
man111 singh
·2014-08-13 23:58:11
