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).

If f(x) = log ((1-x)/(1+x)) -1 lt x lt 1 then show that f(-x) = -f (x)

If f(X) = (ax-b)/(bx-a) , show that f(f(x)) = x

Let f(x)=ln(2+x)-(2x+2)/(x) . Statement I The function f(x) =0 has a unique solution in the domain of f(x). Statement II f(x) is continuous in [a, b] and is strictly monotonic in (a, b), then f has a unique root in (a, b).

Let f(x) = (g(x))/(h(x)) , where g and h are continuous functions on the open interval (a, b). Which of the following statements is true for a lt x lt b ?

Let the function f: R-{-b}->R-{1} be defined by f(x)=(x+a)/(x+b) , a!=b , then f is one-one but not onto (b) f is onto but not one-one (c) f is both one-one and onto (d) none of these

Let f(x)=x^(3)+ax^(2)+bx+c be the given cubic polynomial and f(x)=0 be the corresponding cubic equation, where a, b, c in R. Now, f'(x)=3x^(2)+2ax+b Let D=4a^(2)-12b=4(a^(2)-3b) be the discriminant of the equation f'(x)=0 . IF D=4(a^(2)-3b)lt0 . Then, a. f(x) has all real roots b. f(x) has one real and two imaginary roots c. f(x) has repeated roots d. None of the above

Let a in R and f : R rarr R be given by f(x)=x^(5)-5x+a , then (a) f(x)=0 has three real roots if a gt 4 (b) f(x)=0 has only one real root if a gt 4 (c) f(x)=0 has three real roots if a lt -4 (d) f(x)=0 has three real roots if -4 lt a lt 4

Let f(x)=x^(3)+ax^(2)+bx+c be the given cubic polynomial and f(x)=0 be the corresponding cubic equation, where a, b, c in R. Now, f'(x)=3x^(2)+2ax+b Let D=4a^(2)-12b=4(a^(2)-3b) be the discriminant of the equation f'(x)=0 . If D=4(a^(2)-3b)gt0 and f(x_(1)).f(x_(2)) lt0" where " x_(1),x_(2) are the roots of f(x), then a. f(x) has all real and distinct roots b. f(x) has three real roots but one of the roots would be repeated c. f(x) would have just one real root d. None of the above

Let f(x)=x^(3)+ax^(2)+bx+c be the given cubic polynomial and f(x)=0 be the corresponding cubic equation, where a, b, c in R. Now, f'(x)=3x^(2)+2ax+b Let D=4a^(2)-12b=4(a^(2)-3b) be the discriminant of the equation f'(x)=0 . If D=4(a^(2)-3b)gt0 and f(x_(1)).f(x_(2))=0" where " x_(1),x_(2) are the roots of f(x) , then

Let f(x)=x^(3)+ax^(2)+bx+c be the given cubic polynomial and f(x)=0 be the corresponding cubic equation, where a, b, c in R. Now, f'(x)=3x^(2)+2ax+b Let D=4a^(2)-12b=4(a^(2)-3b) be the discriminant of the equation f'(x)=0 . If D=4(a^(2)-3b)gt0 and f(x_(1)).f(x_(2))gt0 where x_(1),x_(2) are the roots of f(x), then