If f(a,b)`=(f(b)-f(a))/(b-a)` and
`f(a,b,c)=(f(b,c)-f(a,b))/(c-a)` prove that `f(a,b,c)=|{:(f(a),f(b),f(c)),(1,1,1),(a,b,c):}|-:|{:(1,1,1),(a,b,c),(a^(2),b^(2),c^(2)):}|`.

If f'(c)=(f (b)-f(a))/(b-a) , where f(x) =e^(x), a=0 and b=1 , then: c = ….

If ={(a,1),(b,-2),(c,3)},g={(a,-2),(b,0),(c,1)} then f^(2)+g^(2)=

int_(a)^(b-a)f''(x+a)dx equals (a) f(b-a)-f(a) " " (b) f(b)-f(2a) (c) f'(b)-f'(2a) " " (d) f'(b-a)-f'(2a)

If f((ax+b)/(x-a))=x, then f^(-1)(x)= (a) x (b) (bx+a)/(x-b)( c) f(x)

If a!=1, b!=1, c!=1, f(x)= 1/(1-x) and |(a,1,1),(1,b,1),(1,1,c)|=0 then (A) f(a)+f(b)+f(c)=0 (B) f(a)+f(b)+f(c)=1 (C) f(a)+f(b)+f(c)=-1 (D) f(a)f(b)f(c)=1

If f(x)<0,\ x in (a , b) then at the point C(c ,\ f(c)) on y=f(x) for which F(c) is a maximum, f^(prime)(c) is given by a. f^(prime)(c)=(f(b)-f(a))/(b-1) b. \ f^(prime)(c)=(f(b)-f(a))/(a-b) c. f^(prime)(c)=(2(f(b)-f(a)))/(b-a) d. f^(prime)(c)=0

For every function f (x) which is twice differentiable , these will be good approximation of int_(a)^(b)f(x)dx=((b-a)/(2)){f(a)+f(b)} , for more acutare results for cin(a,b),F( c) = (c-a)/(2)[f(a)-f( c)]+(b-c)/(2)[f(b)-f( c)] When c= (a+b)/(2) int_(a)^(b)f(x)dx=(b-a)/(4){f(a)+f (b)+2 f ( c) }dx Good approximation of int_(0)^(pi//2)sinx dx , is

For every function f (x) which is twice differentiable , these will be good approximation of int_(a)^(b)f(x)dx=((b-a)/(2)){f(a)+f(b)} , for more acutare results for cin(a,b),F( c) = (c-a)/(2)[f(a)-f( c)]+(b-c)/(2)[f(b)-f( c)] When c= (a+b)/(2) int_(a)^(b)f(x)dx=(b-a)/(4){f(a)+f (b)+2 f ( c) }dx If lim_(t toa) (int_(a)^(t)f(x)dx-((t-a))/(2){f(t)+f(a)})/((t-a)^(3))=0 , then degree of polynomial function f (x) atmost is

if cos^(-1)(a)+cos^(-1)(b)+cos^(-1)(c)=3 pi and f(1)=2,f(x+y)=f(x)f(y) for all x,y: then a^(2f(1))+b^(2f(2))+c^(2f(3))+(a+b+c)/(a^(2f(1))+b^(2f(2))+c^(2f(3)))