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 f(x) g(x) and h(x) are three polynomials of degree 2 and Delta = |( f(x), g(x), h(x)), (f'(x), g'(x), h'(x)), (f''(x), g''(x), h''(x))| then Delta(x) is a polynomial of degree (dashes denote the differentiation).

If f(x), g(x), h(x) are polynomials of three degree, then phi(x)=|(f'(x),g'(x),h'(x)), (f''(x),g''(x),h''(x)), (f'''(x),g'''(x),h'''(x))| is a polynomial of degree (where f^n (x) represents nth derivative of f(x))

Evaluate: int(a x^2+b x+c)/((x-a)(x-b)(x-c))\ dx ,\ \ w h e r e\ a ,\ b ,\ c are distinct.

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 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:

A function f satisfies the condition f(x)=f^(prime)(x)+f''(x)+f'''(x)....+infinity ,w h e r ef(x) is a differentiable function indefinitely and dash denotes the order of derivative. If f(0)=1,t h e nf(x) is e^(x/2) (b) e^x (c) e^(2x) (d) e^(4x)

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),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