If`D=d i ag[d_1, d_2, d_n]` , then prove that `f(D)=d i ag[f(d_1),f(d_2), ,f(d_n)],w h e r ef(x)` is a polynomial with scalar coefficient.

If x=1/z and y=f(x), then prove that, (d^2f)/(dx^2)=2z^3dy/dz+z^4(d^2y)/(dz^2)

If a : b = c : d = e : f , then prove that (a^(2) + c^(2) + e^(2)) (b^(2)+d^(2)+f^(2)) = (ab + cd + ef)^(2))

If f(x) = logx, then d/(dx)f(logx) =

If f(x)=|x-2| a n d g(x)=f[f(x)],t h e n g^(prime)(x)= ______ for x>20

A curve y=f(x) passes through point P(1,1) . The normal to the curve at P is a (y-1)+(x-1)=0 . If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, then the equation of the curve is (a) ( b ) (c) y=( d ) e^(( e ) (f) K(( g ) (h) x-1( i ))( j ))( k ) (l) (m) (b) ( n ) (o) y=( p ) e^(( q ) (r) K e (s))( t ) (u) (v) (c) ( d ) (e) y=( f ) e^(( g ) (h) K(( i ) (j) x-2( k ))( l ))( m ) (n) (o) (d) None of these

f(x) is cubic polynomial with f(x)=18 and f(1)=-1 . Also f(x) has local maxima at x=-1 and f^(prime)(x) has local minima at x=0 , then (A) the distance between (-1,2)a n d(af(a)), where x=a is the point of local minima is 2sqrt(5) (B) f(x) is increasing for x in [1,2sqrt(5]) (C) f(x) has local minima at x=1 (D)the value of f(0)=15

If a,b,c,d,be the sides of a quadrilateral and g(x)=f[f{f(x)}], where f(x)=1/(1-x), d^2/(a^2+b^2+c^2) is equal to

If d is the minimum distance between the curves f(x)=e^x a n dg(x)=(log)_e x , then the value of d^6 is

The diameter of the smaller piston of a Hydraulic press is d_1 and the larger piston is d_2 . If the applied force on the smaller piston is F_1 then the value of the force F_2 on the larger piston 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