If the cubic equation `3x^3 +px +5=0` has exactly one real root then show that ` p gt 0`

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

If the equation x^(3) +px +q =0 has three real roots then show that 4p^(3)+ 27q^(2) lt 0 .

If (1+alpha)/(1-alpha),(1+beta)/(1-beta), (1+gamma)/(1-gamma) are the cubic equation f(x) = 0 where alpha,beta,gamma are the roots of the cubic equation 3x^3 - 2x + 5 =0 , then the number of negative real roots of the equation f(x) = 0 is :

If (1+alpha)/(1-alpha),(1+beta)/(1-beta), (1+gamma)/(1-gamma) are the cubic equation f(x) = 0 where alpha,beta,gamma are the roots of the cubic equation 3x^3 - 2x + 5 =0 , then the number of negative real roots of the equation f(x) = 0 is :

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

If 2a+3b+6c = 0, then show that the equation a x^2 + bx + c = 0 has atleast one real root between 0 to 1.

If 2a+3b+6c = 0, then show that the equation a x^2 + bx + c = 0 has atleast one real root between 0 to 1.

find the condition that px^(3) + qx^(2) +rx +s=0 has exactly one real roots, where p ,q ,r,s,in R

If the equation sin ^(2) x - k sin x - 3 = 0 has exactly two distinct real roots in [0, pi] , then find the values of k .