IIT JEE , AIEEE Forum - Mathematics , Trignometry

akhileshc

Q 1.Eqn x^3-3x+a=0 will have exactly one real root if a belong to.... ans:(-inf.,-2)U(2,inf)
Q2.If f(x)=cosx+a^2x+b is an increasing function for all values of x then a belongs to..? ans:(-inf.,-2)U[0,inf]

hsbhatt

First look at f(x) = x^3 - 3x+a

It attains a minimum at x=1 and a maximum at x=-1.

$f(x)_{x=-1} = a+2 ext{and} f(x)_{x=1} = a-2$

Drawing an approximate graph at this point will help. You can see that if either a+2 le 0

or if

, then only one real root exists.

Hence it is necessary and sufficient that a
otin (-2,2)

for the given function to have exactly one real root.

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