If `f(x)=|x|^3` , show that `f"(x)` exists for all real `x` and find it.

If f(x)=|x| then show that f'(3) =1

A function f:R->R satisfies the relation f((x+y)/3)=1/3|f(x)+f(y)+f(0)| for all x,y in R. If f'(0) exists, prove that f'(x) exists for all x, in R.

If f is a real function such that f(x) > 0,f^(prime)(x) is continuous for all real x and a xf^(prime)(x)geq2sqrt(f(x))-2af(x),(a x!=2), show that sqrt(f(x))geq(sqrt(f(1)))/x ,xgeq1 .

If f(x)=(|x-a|)/(x-a) , then show that lim_(xrarra) f(x) does not exist.

If f(x)=(|x|)/(x) , then show that lim_(xrarr0) f(x) does not exist.

Let (f(x+y)-f(x))/2=(f(y)-a)/2+x y for all real xa n dydot If f(x) is differentiable and f^(prime)(0) exists for all real permissible value of a and is equal to sqrt(5a-1-a^2)dot Then f(x) is positive for all real x f(x) is negative for all real x f(x)=0 has real roots Nothing can be said about the sign of f(x)

If f(x)=2 for all real numbers x, then f(x+2)=

If f is real-valued differentiable function such that f(x)f'(x)<0 for all real x, then

If f is real-valued differentiable function such that f(x)f'(x)<0 for all real x, then

Find the values ' a ' for which the function f(x)=(a+2)x^3-3a x^2+9a x-1 decreases for all real values of x .