If `x in R ,a n da ,b ,c` are in ascending or descending order of magnitude, show that `(x-a)(x-c)//(x-b)(w h e r ex!=b)` can assume any real value.

For real x, the function (x-a)(x-b)/(x-c) will assume all real values provided

A certain polynomial P(x)x in R when divided by x-a ,x-ba n dx-c leaves remainders a , b ,a n dc , resepectively. Then find remainder when P(x) is divided by (x-a)(x-b)(x-c)w h e r eab, c are distinct.

Evaluate int_a^b(dx)/(sqrt(x)),w h e r ea , b > 0.

Differentiate tan^(-1)((sqrt(1+x^2)-1)/x)w.r.t tan^(-1)x ,w h e r ex!=0.

Find the largest term in the expansion of (3+2x)^(50),w h e r ex=1//5.

If a ,b ,c in R , then find the number of real roots of the equation =|x c-b-c x a b-a x|=0

Prove that a^4+b^4+c^4> a b c(a+b+c),w h e r ea ,b ,c > 0.

If a ,b ,c in R^+a n d2b=a+c , then check the nature of roots of equation a x^2+2b x+c=0.

Let a and b be two positive real numbers. Then the value of int_(a)^(b)(e^(x//a)-e^(b//x))/(x)dx is

The n t h derivative of x e^x vanishes when (a) x=0 (b) x=-1 (c) x=-n (d) x=n