Thus `f(0)=f(1)` and hence equation `f\'(x)=0` has at least one root between 0 and 1. Show that equation `(x-1)^5+(2x+1)^9+(x+1)^21=0` has exactly one real root.

Show that the equation (x-1)^5+(x+2)^7+(7x-5)^9=10 has exactly one root.

Prove that the equation x^(3) + x – 1 = 0 has exactly one real root.

Find the condition if the equation 3x^(2)+4ax+b=0 has at least one root in (0,1).

Show that the equation x^(5)-3x-1=0 has a unique root in [1,2] .

If the equation x^(2)-kx+1=0 has no real roots then

If f(x)=1+x^(m)(x-1)^(n), m , n in N , then f'(x)=0 has atleast one root in the interval

Let f(x) = 8x^3 - 6x^2 - 2x +1 then, (A) f(x) = 0 has no root in (0,1) (B) f(x) = 0 has at least one roct in (0,1) (C) f'(x) vanishes for some x in (0,1), (D) f(x)=0 has at most two real roots (0, 1)

If f(x)=x ln x-2, then show that f(x)=0 has exactly one root in the interval (1,e).

Statement-1: Let a,b,c be non zero real numbers and f(x)=ax^2+bx+c satisfying int_0^1 (1+cos^8x)f(x)dx=int_0^2(1+cos^8x)f(x)dx then the equation f(x)=0 has at least one root in (0,2) .Statement-2: If int_a^b g(x)dx vanishes and g(x) is continuous then the equation g(x)=0 has at least one real root in (a,b) . (A) Both 1 and 2 are true and 2 is the correct explanation of 1 (B) Both 1 and 2 are true and 2 is not correct explanation of 1 (C) 1 is true but 2 is false (D) 1 is false but 2 is true

If quadratic equation f(x)=x^(2)+ax+1=0 has two positive distinct roots then