If f'(x)=0 for all values of x in an interval, then show that f(x) is constant in that interval.

If f(x)=x ln x-2, then show that f(x)=0 has exactly one root in the interval (1,e).

f(x) = x^(x) is decreasing in the interval

Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f '(x) ne 0 for all x in R . If |[f(x)" "f'(x)], [f'(x)" "f''(x)]|= 0 , for all x in R , then the value of f(1) lies in the interval:

If f(x) = x- sqrt x is increasing in the interval..

The function f(x) = x^x decrease on the interval

If f(x)is integrable function in the interval [-a,a] then show that int_(-a)^(a)f(x)dx=int_(0)^(a)[f(x)+f(-x)]dx.

If a function f(x) increases in the interval (a,b). then the function phi(x)=[f(x)]^(n) increases in the same interval and phi(x)!=f(x) if