`P=x^3-1/x^3, Q=x-1/x` `x in (1,oo)` then minimum value of `P/(sqrt(3)Q^2)`

Let P=x^(3)-(1)/(x^(3)),Q=x-(1)/(x) and a is the minimum value of (P)/(Q^(2)). Then the value of [a] is (where [x] is greatest integer <=x .)

If p(x) = x^3+3 and q(x)= x-3 then find the value of p(x).q(x)

If the roots of the equation x^(2)+(p+1)x+2q-q^(2)+3=0 are real and unequal AA p in R then find the minimum integral value of (q^(2)-2q)

Let P(x) and Q(x) be two quadratic polynomials such that minimum value of P(x) is equal to twice of maximum value of Q(x) and Q(x)>0AA x in(1,3) and Q(x)<0AA x in(-oo ,1)uu(3, oo) If sum of the roots of the equation P(x) =0 is 8 and P(3)-P(4)=1 and distance between the vertex of y=P(x) and vertex of y=Q(x) is 2sqrt(2) The polynomial P(x) is:

Let P(x) and Q(x) be two quadratic polynomials such that minimum value of P(x) is equal to twice of maximum value of Q(x) and Q(x)>0AA x in(1, 3) and Q (x)<0AA x in(-oo, 1)uu(3, oo) . If sum of the roots of the equation P(x)=0 is 8 and P(3)-P(4)=1 and distance between the vertex of y=P(x) and vertex of y=Q(x) is 2sqrt(2) The polynomial P(x) is:

If the solution of the equation |(x^4-9)-(x^2+3)|=|x^4-9|-|x^2+3| is (-oo, p]uu[q ,oo) then value of p+q is (a) p=-3 (b) p=-2 (a) q=2 (d) q=3

If the solution of the equation |(x^(4)-9)-(x^(2)+3)|=|x^(4)-9|-|x^(2)+3| is (-oo,p]uu[q,oo) then value of p+q is (a) p=-3(b)p=-2(a)q=2(d)q=3

f(x)=3^(3)sqrt(x)-5 and g(x)=2px+q^(2) . If f(g(2))=7 ,what is the minimum value of (p+q) if p is a positive integer?

If x=(6pq)/(p+q) , then find the value of (x+3p)/(x-3p) +(x+3q)/(x-3q)