Let `a in R` and let `f: R` be given by `f(x)=x^5-5x+a ,` then
(a) `f(x)` has three real roots if `agt4`
(b)`f(x)` has only one real roots if `a gt4`
(c)`f(x)` has three real roots if `alt-4`
(d)`f(x)` has three real roots if `-4ltalt4`

Find the value of a if x^3-3x+a=0 has three distinct real roots.

Find the value of k if x^3-12x+k=0 has three real distinct roots.

Let f(x)=sinx+a x+bdot Then which of the following is/are true? (a) f(x)=0 has only one real root which is positive if a >1, b (b) f(x)=0 has only one real root which is negative if a >1, b>0. (c) f(x)=0 has only one real root which is negative if a (d) none of these

If f(x)=int(x^8+4)/(x^4-2x^2+2)dx and f(0)=0,t h e n (a) f(x) is an odd function (b) f(x) has range R (c) f(x) has at least one real root (d) f(x) is a monotonic function.

If he equation x^3+a x^2+b x+216=0 has three real roots in G.P., then b//a has the value equal to _____.

Let R be the set of real numbers and f:R to R be defined by f(x)=2x^(2)-5x+6 , then the value of f^(-1)(2) is -

Let R be the set of all real numbers f:RrarrR be given by f(x)=3x^2+1 .Then the set f^(-1) , (1,6) is

f(x) is a polynomial function, f: R rarr R, such that f(2x)=f'(x)f''(x). Equation f(x) = x has (A) three real and positive roots (B) three real and negative roots (C) one real root (D) three real roots such that sum of roots is zero

A quadratic trinomial P(x)=a x^2+b x+c is such that the equation P(x)=x has no real roots. Prove that in this case equation P(P(x))=x has no real roots either.

Let RR be the set of real numbers and f : RR to RR be defined by f(x)=sin x, then the range of f(x) is-