Let `f(x)` is a quadratic expression with positive integral coefficients such that for every `alpha, beta in R, beta lt alpha,int_alpha^beta f(x) dx lt 0`. Let `g(t) = f''(t) f(t), and g(0) = 12`, then

Let f:[0,4]rarr R be a differentiable function then for some alpha,beta in (0,2)int_(0)^(4)f(t)dt

Let int_(alpha)^( beta)(f(alpha+beta-x))/(f(x)+f(alpha+beta-x))dx=4 then

Let f(x) be a differentiable function and f(alpha)=f(beta)=0(alpha< beta), then the interval (alpha,beta)

Let f(x)=ax^(2)+bx+c=0 be a quadratic equation and alpha,beta are its roots if c!=0 then alpha,beta!=0 and f(1/x) =0 is an equation whose , roots are

Let f(x)=ax^3+bx^2+cx+1 has exterma at x=alpha,beta such that alpha beta 0 (c)one positive root if f(alpha) 0 (d) none of these

If alpha and beta are zeros of f(t) = t^(2) - 4t +3 , find the value of alpha ^(4) beta ^(3) + beta ^(4) alpha ^(3) .

Let f(x) be a differentiable function and f(alpha)=f(beta)=0(alpha

Let f (x) =x ^(2) +2x -t ^(2) and f(x)=0 has two root alpha (t ) and beta (t) (alpha lt beta) where t is a real parameter. Let I (t)=int _(alpha ) ^(beta)f (x) dx. If the maximum value of I (t) be lamda and | lamda| =p/q where p and q are relatively prime positive integers. Find the product (pq).