If `g(x)=(f(x))/((x-a)(x-b)(x-c)),w h e r ef(x)` is a polynomial of degree `<3,` then prove that `(dg(x))/(dx)=|1af(a)(x-a)^(-2)1bf(b)(x-b)^(-2)1cf(c)(x-c)^(-2)|/|a^2a1b^2b1c^2c1|`

If g(x)=(f(x))/((x-a)(x-b)(x-c)) ,where f(x) is a polynomial of degree <3 , then intg(x)dx=|[1,a,f(a)log|x-a|],[1,b,f(b)log|x-b|],[1,c,f(c)log|x-c|]|-:|[1,a, a^2],[ 1,b,b^2],[ 1,c,c^2]|+k (dg(x))/(dx)=|[1,a,-f(a)(x-a)^(-2)],[1,b,-f(b)(x-b)^(-2)],[1,c,-f(c)(x-c)^(-2)]|:-|[1,a,a^2],[1,b,b^2],[1,c,c^2]|

If int(tan^(9)x)dx=f(x)+log|cosx|, where f(x) is a polynomial of degree n in tan x, then the value of n is

If f(x),g(x) and h(x) are three polynomials of degree 2, then prove that phi(x) = |[f(x),g(x),h(x)],[f ^ (prime)(x),g^(prime)(x),h^(prime)(x)],[f^(primeprime)(x),g^(primeprime)(x),h^(primeprime)(x)]| is a constant polynomial.

If f(x),g(x)a n dh(x) are three polynomial of degree 2, then prove that phi(x)=|[f(x)g(x)h(x)];[f'(x)g'(x) h '(x)];[f' '(x)g' '(x )h ' '(x)]| is: constant

Let g(x)=f(x)sinx ,w h e r ef(x) is a twice differentiable function on (-oo,oo) such that f'(-pi)=1. The value of |g^''(-pi)| equals __________

Let f(x) be the polynomial of degree n, then Delta^(n-1) f(x) is a polynomial of degree.

If f(x) is an even function and satisfies the relation x^2dotf(x)-2f(1/x)=g(x),w h e r eg(x) is an odd function, then find the value of f(5)dot

Instead of the usual definition of derivative Df(x), if we define a new kind of derivative D^*F(x) by the formula D*f(x)=lim_(h->0)(f^2(x+h)-f^2(x))/h ,w h e r ef^2(x) mean [f(x)]^2 and if f(x)=xlogx ,then D^*f(x)|_(x=e) has the value (A)e (B) 2e (c) 4e (d) none of these

If f^(prime)(x)=g(x)(x-a)^2,w h e r eg(a)!=0,a n dg is continuous at x=a , then (a) f is increasing in the neighbourhood of a if g(a)>0 (b) f is increasing in the neighbourhood of a if g(a) 0 (d) f is decreasing in the neighbourhood of a if g(a)<0

Let F(x)=e^(x),G(x)=e^(-x) and H(x)=G(F(x)) , where x is a real variable. Then (dH)/(dx) at x=0 is