fundamental differential equation analysis

If I say that the function f:R->R has two derivatives, with f(0) = f'(0) = 0 and the absolute value of f"(x) is less than or equal to one, if the absolute value of x is less than or equal to 1. How can I prove that: f(x) <= 1/2 if x <= 1

fundamental differential equation analysis

If I let I be an open interval containing the point x. (x not) and suppose that the function f:I->R has two derivatives. How can I prove that lim as h->0 (f(x.+h) - 2f(x.) + f(x.-h))/ h^2 = f"(x.)

Simple Function Operations

1a.)Is y=x^4 a single- or multi-valued function? b.)Is y=f(x)=x^2+4x an even, odd, or neither function? c.)What is the inverse function of y=x^4 d.)What is the inverse function of (b.),y=x^2+4x? e.)Is the inverse function from (d.), odd, even, or neither?

Derivatives of functions

5. (a) Find derivatives of these functions chosing appropriate methods.. (See attached file)

Metric Space

Let X be a metric space and x0 in X. Define a function f: X --> R (all real numbers) by f(x) = d(x,x0). Show that f is continuous. HINT: Prove the variant of the triangle inequality which says |d(x,z)-d(y,z)|< d(x,y) for any x,y,z in X