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)a n dh(x) are three polynomial of degree 2, then prove that varphi(x)=|f(x)g(x)h(x)f'(x)g'(x h '(x)f' '(x)g' '(x h ' '(x)| is a constant polynomial.

If f(x),g(x)a n dh(x) are three polynomials of degree 2, then prove that varphi(x)=|f(x)g(x)h(x)f^(prime)(x)g^(prime)(x)h^(prime)(x)f^(x)g^(x)h^(x)|i sacon s t a n tpol y nom i a l

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

Let F(x)=f(x)g(x)h(x) for all real x ,w h e r ef(x),g(x),a n dh(x) are differentiable functions. At some point x_0,F^(prime)(x_0)=21 F(x_0),f^(prime)(x_0)=4f(x_0),g^(prime)(x_0)=-7g(x_0), and h^(prime)(x_0)=kh(x_0) . Then k is________

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^n (-pi)| equals __________

Suppose that f(x) isa quadratic expresson positive for all real xdot If g(x)=f(x)+f^(prime)(x)+f^(x), then for any real x(w h e r ef^(prime)(x)a n df^(x) represent 1st and 2nd derivative, respectively). g(x) 0 c. g(x)=0 d. g(x)geq0

If f(x)=x^n ,f(1)+(f^1(1))/1+(f^2(1))/(2!)+(f^n(1))/(n !),w h e r ef^r(x) denotes the rth order derivative of f(x) with respect to x , is

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