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 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)=x|x|, then prove that f^(prime)(x)=2|x|

Evaluate int[f(x)g^(n)(x)-f^(n)(x)g(x)]dx

If g is inverse of f then prove that f''(g(x))=-g''(x)(f'(g(x)))^(3).

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^(prime)(x)>0a n df^(prime)(x) g(f(x-1)) f(g(x+1))>f(g(x-1)) g(f(x+1))

Evaluate: ifintg(x)dx=g(x),t h e nintg(x){f(x)+f^(prime)(x)}dx

If function f satisfies the relation f(x)*f^(prime)(-x)=f(-x)*f^(prime)(x) for all x ,

If f: RrarrR and g:RrarrR are two given functions, then prove that 2min.{f(x)-g(x),0}=f(x)-g(x)-|g(x)-f(x)|

A nonzero polynomial with real coefficient has the property that f(x)=f^(prime)(x)dot f^(prime prime)(x)dot If a is the leading coefficient of f(x), then the value of 1/(2a) is____