`0^@C` or `0^@F` which one is lesser?

Among which of the following pairs of elements, the first one has lesser ionisation energy than the second-

0^@C = ________ ^@F

Let f(x)=sinx+a x+bdot Then which of the following is/are true? (a) f(x)=0 has only one real root which is positive if a >1, b (b) f(x)=0 has only one real root which is negative if a >1, b>0. (c) f(x)=0 has only one real root which is negative if a (d) none of these

If f(x)=x^alphalogx and f(0)=0 , then the value of alpha for which Rolle's theorem can be applied in [0,1] is

Cosider the non-constant differentiable function f of one variable which obeys the relation (f(x))/(f(y))=f(x-y) . If f'(0)=p and f'(5)=q , then f'(-5) is

In which of the following functions is Rolles theorem applicable? (a)f(x)={x ,0lt=x<1 ;0,x=1on[0,1] (b)f(x)={(sinx)/x ,-pilt=x<0 ;0,x=0on[-pi,0) (c)f(x)=(x^2-x-6)/(x-1)on[-2,3] (d)f(x)={(x^3-2x^2-5x+6)/(x-1)ifx!=1,-6ifx=1on[-2,3]

If f''(x)gt0 and f'(1)=0 such that g(x)=f(cot^2x+2cotx+2) , where 0ltxltpi then the interval in which g(x) is decreasing is :

Let f be a continuous function defined onto [0,1] with range [0,1]. Show that there is some c in [0,1] such that f(c)=1-c

If f(x) is a polynomial, show that between two roots of the polynomial equation f(x)=0, there lies at least one root of the polynomial f'(x)=0.

Consider the function f:(-oo,oo)to(-oo,oo) defined by f(x)=(x^2-a)/(x^2+a),a >0, which of the following is not true?(a) maximum value of f is not attained even though f is bounded. (b) f(x) is increasing on (0,oo) and has minimum at x=0 (c) f(x) is decreasing on (-oo,0) and has minimum at x=0. (d) f(x) is increasing on (-oo,oo) and has neither a local maximum nor a local minimum at x=0.