DSE 2013 qn polynomial eqn

[phelps.phan]

QUESTION 21. The function defined by f(x) = x^5 + 7x^3 + 13x -18
(a) may have 5 real roots.
(b) has no real root.
(c) has 3 real roots.
(d) has exactly 1 real root.

Pls help with the approach

[Granpa Simpson]

Answer is option d...use Descartes Rule

[phelps.phan]

Thanks a lot subhayu :)

[kangkan]

Hi phelps ..you can also use rolles theorem..same ans though

[Granpa Simpson]

@Kangkan: R u sure about that...because i m not sure weder the conditions for using Rolles theorem are satisfied or not...it is continuous, but what is the value of a and b s.t f(a)=f(b)..if you can show that there exists such a, b then only u can use Rolle's theorem..isnt it so..??

[kangkan]

Hi...see if there was more than one real root ,there wud be two points a and b such that f(a)=f(b)=0...hence there there wud be atlest one c where f'(x) =o..here f'(c)= 5x^4+21X^2+13..we can see that this function is never zero....hence there is no a and d such that f(a)=f(b)...hence there is definitely only one real roo or less..but its a 5 degree equation.hence one atleast one root has to be real. hence the answer :)

[Granpa Simpson]

okkk you proved it by contradiction..dats absolutely perfect..

[kangkan]

thank you :)

[phelps.phan]

thanks kangkan :)