Prove that the equation, `3x^3 - 6x^2 + 6x + sinx = 0` has exactly one real root.

Prove that the equation,3x^(3)-6x^(2)+6x+sin x=0 has exactly one real root.

Prove that the equation x^3 + 2x^2 +x + 4 = 0 has exactly one root in the open interval (-3,-2).

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

The smallest integral value of a for which the equation x^(3)-x^(2)+ax-a=0 have exactly one real root,is

If the equation a x^2+b x+c=0 has two positive and real roots, then prove that the equation a x^2+(b+6a)x+(c+3b)=0 has at least one positive real root.

If the equation a x^2+b x+c=0 has two positive and real roots, then prove that the equation a x^2+(b+6a)x+(c+3b)=0 has at least one positive real root.

If x = 1 is one root of the equation x^(3) - 6x^(2) + 11x -6 = 0, find the other roots.

Prove that the equation 3x^(5)+15x-18=0 has exactly one real root.

If b^2<2a c , then prove that a x^3+b x^2+c x+d=0 has exactly one real root.

If b^2<2a c , then prove that a x^3+b x^2+c x+d=0 has exactly one real root.