`f(alpha)=f prime(alpha)=f primeprime(alpha)=0,f(beta)=f prime(beta)=f primeprime(beta)=0 and f(x)` is polynomial of degree 6, then

| alpha alpha1 beta F|=(alpha-P)(beta-alpha)

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) 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.

Let f(x)=ax^3+bx^2+cx+1 has exterma at x=alpha,beta such that alpha beta < 0 and f(alpha) f(beta) < 0 f . Then the equation f(x)=0 has three equal real roots one negative root if f(alpha) (a) 0 and f(beta) (b) one positive root if f(alpha) (c) 0 and f(beta) (d) none of these

Let f(x)=ax^3+bx^2+cx+1 has exterma at x=alpha,beta such that alpha beta < 0 and f(alpha) f(beta) < 0 f . Then the equation f(x)=0 has three equal real roots one negative root if f(alpha) (a) 0 and f(beta) (b) one positive root if f(alpha) (c) 0 and f(beta) (d) none of these

If f(x) and g(x) are functions such that f(x + y) = f(x) g(y) + g(x) f(x), then in |(f(alpha),g(alpha),f(alpha+theta)),(f(beta),g(beta),f(beta+theta)), (f(lambda),g(lambda),f(lambda+theta))| is independent of

If f(x) and g(x) are functions such that f(x + y) = f(x) g(y) + g(x) f(y), then in |(f(alpha),g(alpha),f(alpha+theta)),(f(beta),g(beta),f(beta+theta)), (f(lambda),g(lambda),f(lambda+theta))| is independent of

A function f: R->R satisfies sinxcosy(f(2x+2y)-f(2x-2y)=cosxsiny(f(2x+2y)+f(2x-2y))dot If f^(prime)(0)=1/2,t h e n (a) f^(primeprime)(x)=f(x)=0 (b) 4f^(primeprime)(x)+f(x)=0 (c) f^(primeprime)(x)+f(x)=0 (d) 4f^(primeprime)(x)-f(x)=0

A function f: R->R satisfies sinxcosy(f(2x+2y)-f(2x-2y)=cosxsiny(f(2x+2y)+f(2x-2y))dot If f^(prime)(0)=1/2,t h e n (a) f^(primeprime)(x)=f(x)=0 (b) 4f^(primeprime)(x)+f(x)=0 (c) f^(primeprime)(x)+f(x)=0 (d) 4f^(primeprime)(x)-f(x)=0

Repeated roots : If equation f(x) = 0, where f(x) is a polyno- mial function, has roots alpha,alpha,beta,… or alpha root is repreated root, then f(x) = 0 is equivalent to (x-alpha)^(2)(x-beta)…=0, from which we can conclude that f(x)=0 or 2(x-alpha)[(x-beta)...]+(x-alpha)^(2)[(x-beta)...]'=0 or (x-alpha) [2 {(x-beta)...}+(x-alpha){(x-beta)...}']=0 has root alpha . Thus, if alpha root occurs twice in the, equation, then it is common in equations f(x) = 0 and f'(x) = 0. Similarly, if alpha root occurs thrice in equation, then it is common in the equations f(x)=0, f'(x)=0, and f'''(x)=0. If alpha root occurs p times and beta root occurs q times in polynomial equation f(x)=0 of degree n(1ltp,qltn) , then which of the following is not ture [where f^(r)(x) represents rth derivative of f(x) w.r.t x] ?