Let `f(x)=a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x ,` where `a_i ' s` are real and `f(x)=0` has a positive root `alpha_0dot` Then `f^(prime)(x)=0` has a positive root `alpha_1` such that `0

f(x)=a_(5)x^(5)+a_(4)x^(4)+a_(3)x^(3)+a_(2)x^(2)+a_(1)x where a_(i)^( prime)s are real and f(x)=0 has a positive root alpha_(0). Then f'(x)=0 has a positive root alpha_(1) such that 0

Let f(x)=ax^(5)+bx^(4)+cx^(3)+dx^(2)+ ex , where a,b,c,d,e in R and f(x)=0 has a positive root. alpha . Then,

Let F(x)=1+f(x)+(f(x))^(2)+(f(x))^(3) where f(x) is an increasing differentiable function and F(x)=0 hasa positive root then

Prove that x^(6)+x^(4)+x^(2)+x+3=0 has no positive real roots.

If f(x) is a real valued polynomial and f (x) = 0 has real and distinct roots, show that the (f'(x)^(2)-f(x)f''(x))=0 can not have real roots.

Let f(x)=x^3+2x^2+x+5. Show that f(x)=0 has only one real root alpha such that [alpha]=-3

Let f(x)=ax^3+bx^2+cx+1 has exterma at x=alpha,beta such that alpha beta < 0 and f(alpha) f(beta) < 0 f . Then the equation f(x)=0 has three equal real roots one negative root if f(alpha) (a) 0 and f(beta) (b) one positive root if f(alpha) (c) 0 and f(beta) (d) none of these

Let f(x)=x^(2)+ax+b, where a,b in R. If f(x)=0 has all its roots imaginary then the roots of f(x)+f'(x)+f(x)=0 are

If f(x) be a quadratic polynomial such that f(x)=0 has a root 3 and f(2)+f(-1)=0 then other root lies in

Let f(x)=ax^3+bx^2+x+d has local extrema at x=alpha and beta such that alphabetalt0 and f(alpha).f(beta)gt0 . Then the equation f(x)=0 (A) has 3 distinct real roots (B) has only one real which is positive o a.f(alpha)lt0 (C) has only one real root, which is negative a.f(beta)gt0 (D) has 3 equal roots