If for some differentiable function `f(alpha)=0a n df^(prime)(alpha)=0,` Statement 1: Then sign of `f(x)` does not change in the neighbourhood of `x=alpha` Statement 2: `alpha` is repeated root of `f(x)=0`

Suppose that f(0)=0a n df^(prime)(0)=2, and let g(x)=f(-x+f(f(x)))dot The value of g' (0) is equal to _____

If f((x+2y)/3)=(f(x)+2f(y))/3AAx ,y in Ra n df^(prime)(0)=1,f(0)=2, then find f(x)dot

If f^(prime)x=g(x)(x-a)^2,w h e r eg(a)!=0,a n dg is continuous at x=a , then f is increasing in the neighbourhood of a if g(a)>0 f is increasing in the neighbourhood of a if g(a) 0 f is decreasing in the neighbourhood of a if g(a)<0

Let f(x) be defined as f(x)={tan^(-1)alpha-5x^2,0ltxlt1 and -6x ,xgeq1 if f(x) has a maximum at x=1, then find the values of alpha .

Let f(x0 be a non-constant thrice differentiable function defined on (-oo,oo) such that f(x)=f(6-x)a n df^(prime)(0)=0=f^(prime)(x)^2=f(5)dot If n is the minimum number of roots of (f^(prime)(x)^2+f^(prime)(x)f^(x)=0 in the interval [0,6], then the value of n/2 is___

Let alpha,beta are the roots of x^2+b x+1=0. Then find the equation whose roots are - (alpha+1//beta)and-(beta+1//alpha) .

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] ?

For every twice differentiable function f: R->[-2,\ 2] with (f(0))^2+(f^(prime)(0))^2=85 , which of the following statement(s) is (are) TRUE? There exist r ,\ s in R where r oo)f(x)=1 (d) There exists alpha in (-4,\ 4) such that f(alpha)+f"(alpha)=0 and f^(prime)(alpha)!=0