[" (c) "(-c,e)],[" 1.If "O/(x)" is continuous at "x=alpha" such that "phi(alpha)<0" and "f(x)],[" is a function such that "],[f'(x)=(ax-a^(2)-x^(2))phi(x)" for all "x," then "f(x)" is "],[" (a) increasing in the neighbourhood of "x=alpha],[" (b) decreasing in the neighbourhood of "x=alpha],[" (c) constant in the neighbourhood of "x=alpha],[" (d) minimum at "x=alpha]

If phi (x) is continuous at x= alpha such that f(x)=(ax-a^2-x^2)phi(x) for all x, then f(x) is

If f(x)=(tan x cot alpha)^((1)/(x-a)),x!=alpha is continuous at x=alpha, then the value of f(alpha) is

Let f : R rarr R be defined as f(x) = {{:((x^(3))/((1-cos 2x)^(2)) log_(e)((1+2x e^(-2x))/((1-x e^(-x))^(2))),",",x ne 0),(alpha,",", x =0):} . If f is continuous at x = 0, then alpha is equal to :

If alpha, beta are the roots of ax^(2)+bx+c=0 and f(x) is continuous at x=alpha , where f(x)=(1-cos (ax^(2)+bx+c))/((x-alpha)^(2)) , for x != alpha , then f(alpha)=

If the functions f : R -> R and g : R -> R are such that f(x) is continuous at x = alpha and f(alpha)=a and g(x) is discontinuous at x = a but g(f(x)) is continuous at x = alpha. where, f(x) and g(x) are non-constant functions (a) x=alpha extremum of f(x) and x=alpha is an extremum g(x) (b) x=alpha may not be extremum f(x) and x=alpha is an extermum of g(x) (c) x=alpha is an extremum of f(x)and x=alpha may not be an extremum g(x) (d) not of the above

Let f and g be two real functions; continuous at x=a Then alpha f and (1)/(f) is continuous at x=a provided f(a)!=0 and alpha is a real number

Let f (x) = log _e (sinx ), ( 0 lt x lt pi ) and g(x) = sin ^(-1) (e ^(-x)), (x ge 0) . If alpha is a positive real number such that a = ( fog)' ( alpha ) and b = (fog ) ( alpha ) , then (A) aalpha ^(2) - b alpha - a = 0 (B) a alpha ^(2) - b alpha - a = 1 (C) a alpha ^(2) + b alpha - a = - 2 alpha ^(2) (D) a alpha ^(2) + b alpha + a = 0

alpha beta x ^(alpha-1) e^(-beta x ^(alpha))

If the roots of a_(1)x^(2)+b_(1)x+c_(1)=0 are alpha_(1),beta_(1) and those of a_(2)x^(2)+b_(2)x+c_(2)=0 are alpha_(2),beta_(2) such that alpha_(1)alpha_(2)=beta_(1)beta_(2)=1 then

If inte^(x)sin x dx=(sqrt2)/(2)e^(x)sin(x+alpha)+c, then alpha=