If `f(x)=|(x+c_1,x+a,x+a),(x+b,x+c_2,x+a),(x+b,x+b,x+c_3)|` then show that f(x) is linear in x. Hence deduce`f(0)= (bg(a)-ag(b))/((b-a))` where `g(x)=(c_1-x)(c_2-x)(c_3-x)`

If f(x)={:abs((x+a^(2),ab,ac),(ab,x+b^(2),bc),(ac,bc,x+c^(2))):} , then find f'(x).

Let f(x)=(x-a)(x-b)(x-c), a lt b lt c. Show that f'(x)=0 has two roots one belonging to (a, b) and other belonging to (b, c).

Let f(x) = a x^2 + bx + c , where a, b, c in R, a!=0 . Suppose |f(x)| leq1, x in [0,1] , then

If f(x) = 3(2x + 3)^(2//3) + 2x + 3 , then: (a) f(x) is continuous but not differentiable at x = - (3)/(2) (b) f(x) is differentiable at x = 0 (c) f(x) is continuous at x = 0 (d) f(x) is differentiable but not continuous at x = - (3)/(2)

If f(x) = {{:(x - 3",",x lt 0),(x^(2) - 3x + 2",",x ge 0):} , then g(x) = f(|x|) is a. g'(0^(+)) = - 3 b. g'(0^(-)) = -3 c. g'(0^(+)) = g'(0^(-)) d. g(x) is not continuous at x = 0

If g(x)=(f(x))/((x-a)(x-b)(x-c)), where f(x) is a polynomial of degree lt3, then prove that (dg(x))/(dx)=|{:(1,a,f(a)(x-a)^(-2)),(1,b,f(b)(x-b)^(-2)),(1,c,f(c)(x-c)^(-2)):} |divide|{:(a^(2),a,1),(b^(2),b,1),(c^(2),c,1):}|.

If f(x)=|((x-a)^(4),(x-a)^(3),1),((x-b)^(4),(x-b)^(3),1),((x-c)^(4),(x-c)^(3),1)|" then "f'(x)=lambda.|((x-a)^(4),(x-a)^(2),1),((x-b)^(4),(x-b)^(2),1),((x-c)^(4),(x-c)^(2),1)| . The value of lambda is

If f(x)=ax+b and g(x)=cx+d, then f(g(x))=g(f(x)) is equivalent to (a) f(a) = g(c ) (b) f(b) = g(b) (c) f(d) = g(b) (d) f(c ) = g(a)