Let `f(x)=sinx+a x+bdot` Then which of the following is/are true? `f(x)=0` has only one real root which is positive if `a >1, b<0.` `f(x)=0` has only one real root which is negative if `a >1, b<0.` `f(x)=0` has only one real root which is negative if `a >1, b > 0.` `non eoft h e s e`

Let f(x)=sinx+ax+b .Then which of the following is/are true?

Let f be a real-valued function defined on interval (0,oo) ,by f(x)=lnx+int_0^xsqrt(1+sint).dt . Then which of the following statement(s) is (are) true? (A). f"(x) exists for all in (0,oo) . " " (B). f'(x) exists for all x in (0,oo) and f' is continuous on (0,oo) , but not differentiable on (0,oo) . " " (C). there exists alpha>1 such that |f'(x)|<|f(x)| for all x in (alpha,oo) . " " (D). there exists beta>1 such that |f(x)|+|f'(x)|<=beta for all x in (0,oo) .

If the equation x^2=a x+b=0 has distinct real roots and x^2+a|x|+b=0 has only one real root, then which of the following is true? b=0, a >0 b. b=0, a 0, a >0

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

If f(x)a n dg(x) are two positive and increasing functions, then which of the following is not always true? [f(x)]^(g(x)) is always increasing [f(x)]^(g(x)) is decreasing, when f(x) 1. If f(x)>1,t h e n[f(x)]^(g(x)) is increasing.

Let f(x)=a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x , where a_i ' s are real and f(x)=0 has a positive root alpha_0dot Then f^(prime)(x)=0 has a positive root alpha_1 such that 0

If both the distinct roots of the equation |sinx|^2+|sinx|+b=0in[0,pi] are real, then the values of b are [-2,0] (b) (-2,0) [-2,0] (d) non eoft h e s e

The range of the following function is f(x)=sqrt((1-cosx)sqrt((1-cosx)sqrt((1-cosx)sqrt(oo)))) (a)(0,1) (b) (0,1/2) (c) (0,2) (d) non eoft h e s e

If f(x)={{:(x^(3)(1-x)",",xle0),(xlog_(e)x+3x",",xgt0):} then which of the following is not true?

If a x^2+(b-c)x+a-b-c=0 has unequal real roots for all c in R ,t h e n