Let `f(x)=(x-a)(x-b)(x-c), a lt b lt c.` Show that `f'(x)=0` has two roots one belonging to (a, b) and other belonging to (b, c).

Let f: RvecR ,f(x)=(x-a)/((x-b)(x-c)),b > cdot If f is onto, then prove that a in (b , c)dot

If x^(2)+bx+x=0 has no real roots and a+b+c lt 0 then

Let f (x) =x ^(2)-bx+c,b is an odd positive integer. Given that f (x)=0 has two prime numbers an roots and b+c =35. If the least value of f (x) AA x in R is lamda, then |(lamda)/(3)| is equal to (where [.] denotes greatest integer functio)

Let f (x)= (x-a) (x-b)(x-c) be a ral vlued function where a lt b c (a,b,c in R) such that f ''(alpha ) =0. Then if alpha in ( c _(1), c _(2)), which one of the following is correct ?

Let f(x)=(x-a)^2+(x-b)^2+(x-c)^2dot Then, f(x) has a minimum at x= (a+b+c)/3 (b) 1/2 (c) 1/8 (d) none of these

Let f(x)=x^2-bx+c,b is an odd positive integer. Given that f(x)=0 has two prime numbers as roots and b+c=35. If the least value of f(x) AA x in R" is "lambda," then "[|lambda/3|] is equal to (where [.] denotes greatest integer function)

If (x-a) (x-b) lt 0 , then x lt a, and x gt b .

Let f(x)=x^(2)+ax+b , where a, b in R . If f(x)=0 has all its roots imaginary, then the roots of f(x)+f'(x)+f''(x)=0 are

Let f (x) =ax ^(2) +bx + c,a ne 0, such the f (-1-x)=f (-1+ x) AA x in R. Also given that f (x) =0 has no real roots and 4a + b gt 0. Let p =b-4a, q=2a +b, then pq is:

If a twice differentiable function f(x) on (a,b) and continuous on [a, b] is such that f''(x)lt0 for all x in (a,b) then for any c in (a,b),(f(c)-f(a))/(f(b)-f(c))gt