Prove that f(x)=`a^x`, `0`<`a`<`1`, is decreasing in R.`

Prove that f(x) = {sinx/x ; x != 0 and 1 ; x=0 . is continuous at x=0 .

Prove that f(x)=tan ^(-1)(sin x + cos x ),x gt 0 f(0,pi/4) is increasing function

Prove that f(x)=x-sinx is an increasing function.

If (x)=tan x , the prove that :f(x)+f(-x)=0

Prove that f(x)={((x-|x|)/x, x!=0),( 2 , x=0):} is discontinuous at x=0 .

If f(x)=a x^2+b x+c ,g(x)=-a x^2+b x+c ,where ac !=0, then prove that f(x)g(x)=0 has at least two real roots.

If f(x) = (a-x^n)^(1/n), a > 0 and n in N , then prove that f(f(x)) = x for all x.

Prove that f(x)=sqrt(|x|-x) is continuous for all xgeq0.

Prove that f(x)=sqrt(|x|-x) is continuous for all xgeq0 .

lf f'(x) > 0,f"(x)>0AA x in (0,1) and f(0)=0,f(1)=1 ,then prove that f(x)f^-1(x) lt x^2AA x in (0,1)